m 


WORKS  BY 

L.  A.  WATERBURY 

PUBLISHED   BY 

JOHN  WILEY  &  SONS,  Inc. 

Stresses  in  Structural  Steel 
Angles  with  Special  Tables 

v  +  77  pages.  51  x  8.  Illus- 
trated. $1.25  net. 

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STRESSES 


IN 


STRUCTURAL  STEEL  ANGLES 

WITH   SPECIAL  TABLES 


BY 

L.  A.  WATERBURY,  C.E. 

M.  AM.  Soc.  C.E. 

PROFESSOR  OF  CIVIL  AND  ARCHITECTURAL  ENGINEERING, 
UNIVERSITY  OF  ARIZONA 


FIRST. EDITION 


NEW  YORK 

JOHN   WILEY   &   SONS,    INC. 

LONDON  :   CHAPMAN  &  HALL,   LIMITED 
1917 


Copyright,  1917 

BY 
L.  A.  WATERBURY 


PRESS  OF 

BRAUNWORTH    &   CO. 

BOOK   MANUFACTURERS 

BROOKLYI      N.   V. 


PREFACE 


ALTHOUGH  steel  angles  are  extensively  used  in  structural 
framing,  there  are  important  factors  affecting  the  stresses 
in  such  members  that  are  not  commonly  considered  in  analyses 
which  are  made  for  purposes  of  design.  This  volume  has 
been  prepared  in  the  hope  that  it  may  serve  to  indicate  the 
nature  of  some  of  these  factors,  and  that  it  may  furnish  the 
means  for  their  consideration  in  practical  problems. 

On  account  of  lack  of  symmetry  of  the  section,  the  product 
of  inertia  is  involved  in  the  computation  of  bending  stresses, 
but  the  ordinary  structural  handbooks  do  not  include  values 
of  this  element.  The  author  has  computed  the  values  of 
the  products  of  inertia  for  commercial  angles  2  by  2  inches 
and  larger,  which  values  are  included  hi  Table  I  and  Table  II 
at  the  back  of  the  book. 

The  section  modulus  polygon,  which  is  a  very  useful 
device  for  studying  and  for  determining  the  flexural  stresses 
in  unsymmetrical  sections,  is  explained,  and  for  commercial 
angles  2  by  2  inches  and  larger  coordinates  for  the  vertices 
of  the  polygons  are  given  in  Table  III. 

The  subject  of  end  connections  and  their  efficiency  is 
considered,  and  in  Table  IV  and  Table  V  are  given  a  consid- 
erable number  of  values  of  efficiency,  of  equivalent  effective 
area,  and  of  total  tension  allowable  for  a  maximum  unit 
stress  of  16,000  pounds  per  square  inch,  computed  in  accord- 
ance with  the  method  which  is  outlined  in  the  text. 

L.  A.  W. 

TUCSON,  ARIZONA, 
January  i,  1917. 

359875 


TABLE  OF  CONTENTS 


THEORY   AND   DISCUSSION 

ARTICLE  PAGE 

1.  Relation  between  Bending  Moment  and  Flexural  Stress i 

2.  Expressions  for  the  Section  Modulus 2 

3.  Product  of  Inertia 5 

4.  Section  Modulus  Polygons 6 

5.  Neutral  Axis 9 

6.  Plane  of  Loading 12 

7.  Combined  Stresses 13 

8.  Flexure  for  Angles  in  Pairs 15 

9.  Transfer  of  Stress  by  Shear  to  an  Outstanding  Leg 17 

10.  Efficiency  of  End  Connections 19 

TABLES 

TABLE 

I.  Elements  of  Angles  with  Equal  Legs 31 

n.  Elements  of  Angles  with  Unequal  Legs 34 

III.  Coordinates  of  Section  Modulus  Polygons  for  Angles 44 

IV.  Efficiency  and  Allowable  Tension  for  Angles  Riveted  through 

One  Leg  to  a  Rigid  Connection  Plate  with  Two  Lines  of 

Rivets 50 

V.  Efficiency  and  Allowable  Tension  for  Angles  Riveted  through 
One  Leg  to  a  Rigid  Connection  Plate  with  One  Line  of 

Rivets 60 

v 


STRESSES  IN 
STRUCTURAL   STEEL  ANGLES 


THEORY  AND   DISCUSSION 

Art.   1.    Relation  between    Bending  Moment  and 
Flexural  Stress  * 

FOR  a  s^Tnmetrical  section,  with  the  plane  of  loading  coin- 
cident with  one  of  the  principal  axes  of  the  section,  the  relation 
between  the  bending  moment  and  the  bending  stress  at  an 
extreme  fiber  is 


in  which    /=unit  stress  at  the  extreme  fiber,  due  to  bending; 
M  =  bending  moment  at  the  section  under  consid- 

eration ; 

/  =  moment  of  inertia  of  the  section; 
y  =  distance  from  extreme  fiber  to  a  line  through 
the  center  of  gravity  of  the  section,  the  line 
being  taken  coincident  with  or  parallel  to 
the  neutral  axis  of  the  section. 

The  value  of  -  is  a  constant  for  any  given  section  and  is 
ordinarily  called  the  section  modulus  or  section  factor.     If 

*  For  an  excellent  treatment  of  this  subject,  see  "  An  Analysis  of  General 
Flexure  in  a  Straight  Bar  of  Uniform  Cross  Section,"  by  Professor  L.  J.  John- 
son, Trans.  Am.  Soc.  C.  E.,  Vol.  LVI,  p.  169,  1906. 


2  „.   STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

the  letter  s  is  used  to  designate  the  section  modulus,  Eq.  (i) 
becomes 


>=?• 


(2) 


I  . 


If  -  is  retained  as  the  value  of  the  section  modulus.  Eq. 

y 

(2)  will  be  applicable  only  for  those  conditions  which  obtained 
for  Eq.  (i),  viz.,  for  symmetrical  sections,  with  the  planes  of 
loading  coincident  with  principal  axes  of  the  sections  con- 
sidered. However,  a  more  general  value  of  the  section  mod- 
ulus can  be  derived,  which,  if  used  in  Eq.  (2),  as  the  value 
of  s,  will  make  this  equation  applicable  to  any  section,  whether 
symmetrical  or  unsymmetrical,  and  for  a  plane  of  loading 
extending  in  any  direction  from  the  center  of  gravity  of  the 
section.  The  term  section  modulus,  unless  otherwise  spe- 
cifically stated,  will  be  used  to  designate  the  general  value, 
and  for  any  case  of  pure  flexure  the  extreme  fiber  stress  will 
then  be  equal  to  the  bending  moment  at  the  section  divided 
by  the  section  modulus. 

Art.  2.    Expressions  for  the  Section  Modulus 

For  any  given  section  subject  to  bending  stresses,  let  the 

rectangular  axes  of  reference, 
X-X  and  Y-Y,  be  taken  with 
their  origin  at  0,  the  center  of 
gravity  of  the  section,  as  indicated 
in  Fig.  i.  Let  OP  be  the  plane 
of  loading  and  nn  the  neutral 
axis.  (It  is  frequently  assumed 
that  the  neutral  axis  is  perpen- 
dicular to  the  plane  of  loading, 
but  this  is  not  true  for  the  general 
case.)  Then  the  general  expression  for  the  section  modulus  is 


cZA 


(Iv  sin  B—J  cos  6)y+(Ix  cos  0—J  sin  6)x 


(3) 


THEORY  AND  DISCUSSION  3 

in  which      s  =  section  modulus  of  the  section  for  the  partic- 

ular plane  of  loading;  . 

7X=  moment  of  inertia  of  the  section  about  X-X; 
Iv  =  moment  of  inertia  of  the  section  about  F-F; 
/  =  product  of  inertia  of  the  section  for  axes  X-X, 

7-7; 

x,  y  =  coordinates  of  the  extreme  fiber; 
0  =  angle  from  OX  to  OP  (see  Fig.  i). 

For  certain  special  cases  Eq.  (3)  can  be  simplified.  Thus, 
for  any  section  which  is  symmetrical  about  either  axis,  J  be- 
comes zero,  and  the  expression  for  the  section  modulus 
becomes 


Iv  sin  0-y+Ix  cos  6-x 


For  a  symmetrical  section  for  which  the  plane  of  loading 
coincides  with  F-F,  6  becomes  90  degrees  and  the  expression 
for  the  section  modulus  becomes 

'  '   ;  -7  ........   (s) 

For  a  symmetrical  section  for  which  the  plane  of  loading 
coincides  with  X—Xt  6  becomes  zero  and  the  expression  for  the 
section  modulus  becomes 


(6) 


In  order  to  derive  the  expression  for  the  section  modulus 
which  is  given  in  Eq.  (3),  let  the  section  be  indicated  by  Fig.  i, 
a/,  y'  being  the  coordinates  of  any  infinitesimal  area  (dA), 
and  a  being  the  angle  which  the  neutral  axis  (nn)  makes  with 
X-X.  M  is  the  bending  moment  for  which  the  plane  of 
action  is  OP,  and  the  other  quantities  are  as  defined  under 
Eq.  (3). 

For  a  combination  of  bending  and  direct  stress  the  neutral 


4  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

axis  will  not  pass  through  the  center  of  gravity  of  the  section, 
but  since  the  resultant  stress  can  in  this  case  be  obtained  by 
combining  the  stresses  obtained  for  pure  flexure  and  for 
axial  loading,  the  derivation  of  the  section  modulus  will  be 
made  for  the  case  of  pure  flexure,  for  which  the  neutral  axis 
will  pass  through  the  center  of  gravity. 

For  any  point  for  which  the  coordinates  are  x1 ',  y',  the 
distance  of  the  point  from  the  neutral  axis  is  (/  cos  a  —  x'  sin  a). 
Therefore,  if  the  stress  is  proportional  to  the  distance  from 
the  neutral  axis, 

f    yf  COSCE  —  x'  sin  a  ,  ^ 

f      y  cos  a  —  x  sin  a  ' 

in  which  /'  =  unit  stress  at  the  point  for  which  the  coordinates 

area/,  /; 

/=unit  stress  at  a  point  for  which  the  coordinates 
are  x,  y,  or  since  x,  y,  are  the  coordinates  the 
extreme  fiber,  /  is  the  extreme  fiber  stress. 

The  bending  moment  may  be  resolved  into  two  com- 
ponents, one  of  which  acts  in  the  plane  of  OF  and  the  other 
in  the  plane  of  OX.  Also,  for  equilibrium,  the  component 
of  the  moment  which  acts  in  the  plane  of  OF  must  be  equal 
to  the  sum  of  the  moments  of  the  stresses  about  the  axis  X-X, 
and  the  component  of  the  moment  in  the  plane  of  OX  must 
be  equal  to  the  sum  of  the  moments  of  the  stresses  about  the 
axis  F-F,  or  using  for  any  stress  the  value  of  /'  obtained 
from  Eq.  (7), 

Msin  8=Cfy'dA 

= '- — : —  Cy'(y'cosa-x'sma)dA,         (8) 

y  cos  a  —  x  smaj 

and 
Mcos  0=  Cf'x'dA 

== 1 —  |  yf(yf  Cos  a  -  x'  sin  a)dA .     .     (o) 

y  cos  a  —  x  sin  a  J 


THEORY  AND  DISCUSSION  5 

Eqs.  (8)  and  (9)  may  be  simplified  by  substituting  for 
C(y')2dA  its  equivalent  Ix,  for  J  (x')2dA,  its  equivalent  Iv,  and 
for  (x'y'dA  its  equivalent  /.  Then,  by  equating  the  value 
of /obtained  from  Eq.  (8)  to  the  value  obtained  from  Eq.  (9), 

sin  0  _  cos  e  ,     . 

Ix  cos  a  —  J  sin  a     J  cos  a  —  Iy  sin  a 
from  which 


By  substituting  the  value  of  tan  a  in  Eq.  (9),  solving  for  —- 

and  by  substituting  for  —  its  equivalent  s,  the  value  obtained 
for  the  section  modulus  is  that  which  is  given  in  Eq.  (3). 

Art.  3.    Product  of  Inertia 

The  product  of  inertia  (/),  which  is  involved  in  the  gen- 
eral equation  for  the  section  modulus,  may  be  defined  by  the 
mathematical  expression, 

J=f x'y'dA, (12) 

in  which  x',  y',  are  the  coordinates  of  any  infinitesimal  area 
(dA). 

Also 

J=Jcg+Akk, (13) 

in  which      /  =  product  of  inertia  for  any  given  rectangular 

axes; 
Jcg  =  product  of  inertia  for  parallel  axes  through  the 

center  of  gravity; 
A  =  area  of  the  section ; 

k,  h  =  coordinates  of  the  center  of  gravity  with  ref- 
erence to  the  given  axes  for  which  /  is 
desired. 


STRESSES  IN  STRUCTURAL  STEEL  ANGLES 


For  any  symmetrical  section  Jcg  is  zero,  or  for  such  a 

section  /  becomes  Akh. 

The  product  of  inertia  may  be 
either  a  positive  or  a  negative 
quantity.  For  an  angle  the  value 
of  /  will  be  positive  for  axes  taken 
as  shown  at  a  and  d  of  Fig.  2, 
and  the  value  will  be  negative 
-  for  axes,  as  shown  at  b  and  c  of 
the  same  figure. 

The  numerical  values  of  J  for 
structural  steel  angles  are  given 
in  the  tables  at  the  back  of  the  book,  to  which  the  proper  sign 
must  be  applied  for  each  particular  case. 

Art.  4.    Section  Modulus  Polygons 

If,  for  any  given  section,  the  values  of  the  section  modulus 
are  computed  for  values  of  d  from  o  to  360  degrees,  and  if 

ab 


FIG.  2. 


FIG.  3. — Types  of  lection  Modulus  Polygons. 

these  values  are  plotted  as  radii  vectors,  the  outline  of  the 
resulting  figure  will  be  a  polygon,  having  a  side  for  each  salient 
angle  of  the  polygon  which  bounds  the  section.  For  some 
common  sections  the  general  forms  of  the  corresponding 
section  modulus  polygons  are  indicated  in  Fig.  3. 


THEORY  AND  DISCUSSION  7 

Having  the  section  modulus  polygon  for  a  given  section, 
the  value  of  the  section  modulus  may  be  obtained  by  scaling 
the  radius  vector  corresponding  to  the  plane  of  loading.  For 
example,  for  the  rectangular  section  indicated  at  a  of  Fig.  3, 
the  vector  OK  is  the  graphical  representation  of  the  section 
modulus  for  the  plane  of  loading  OP,  and  the  bending  stress 
at  the  vertex  B  is  obtained  by  dividing  the  bending  moment 
by  this  value  of  the  section  modulus.  It  will  be  observed 
that  by  using  the  section  modulus  polygon,  not  only  is  the 
required  section  modulus  readily  obtained,  but  also,  the  lines 
of  the  polygon  which  are  intersected  by  the  line  of  direc- 
tion of  OP  indicate  the  points  of  the  section  at  which  the 
critical  bending  stresses  will  occur. 

In  order  to  plot  the  section  modulus  polygon  for  a  par- 
ticular section,  the  coordinates  of  each  vertex  of  the  polygon 
may  be  determined,  and  the  polygon  can  then  be  drawn  by 
connecting  each  two  successive  vertices  with  a  straight  line. 
If  xaj  ya  are  the  coordinates  of  a  and  x6,  yb,  the  coordinates  of 
b,  two  successive  vertices  of  the  polygon  which  bounds  the 
section,  the  coordinates  xab,  yab  of  the  vertex  of  the  section 
modulus  polygon  corresponding  to  the  side  ab  of  the  section 
will  be, 


_  (xa  -  xb)J  -  (ya  -  yb)I 


(Xa-Xb)Ix-(ya-yb)J 

- 


If  ab  is  parallel  to  X-X, 

J 


(16) 


8  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

If  ab  is  parallel  to  F-F. 

x      I, 

Xa 

=  J_ 

Xa 


(18) 
(19) 


cd 


de 


FIG.  4. 


An  inspection  of  Eqs.   (14)   to   (19)   will  show  that  for 
many  cases  the  vertices  of  the  section  modulus  polygon  can 

be  easily  located,  since  the  values  of  -  and  -  can  be  obtained 

y         x 

from  handbooks,  for  structural  sections,  and  since  /  is  zero 
for  symmetrical  sections,  which  includes  I-beams,  channels, 


THEORY  AND  DISCUSSION  9 

and  tees.  For  angles,  Eqs.  (14)  and  (15)  may  be  used,  but 
in  order  that  the  users  of  this  book  may  avoid  this  labor,  the 
coordinates  have  been  computed  and  are  tabulated  at  the 
back  of  the  book  for  the  commercial  sizes  of  steel  angles,  and 
for  variations  in  thickness  of  J  inch.  For  thicknesses  in  odd 
numbers  of  sixteenths  of  an  inch,  the  values  of  the  coordinates 
may  be  obtained  by  interpolation. 

A  typical  section  modulus  polygon  for  an  angle  is  shown 
in  Fig.  4.  If  OP  represents  the  direction  of  the  plane  of 
loading,  the  two  points  at  which  critical  stresses  will  occur 
are  b  and  d  of  the  angle,  but  if  the  plane  of  loading  is  shifted 
to  OP'  the  points  of  critical  stresses  will  be  a  and  d. 

The  most  advantageous  plane  of  loading  is  that  for 
which  there  is  the  greatest  sec- 
tion modulus.  For  an  angle  used 
as  a  purlin,  for  a  vertical  load, 
a  study  of  Fig.  4  will  show  that 
the  purlin  should  be  set  as  indi- 
cated at  a  of  Fig.  5,  not  as  shown 

r  IG.   O. 

at  b. 

It  will  be  observed  that  for  an  angle  in  tension  or  com- 
pression, the  application  of  the  load  at  the  gauge  line  is  not 
far  from  the  most  advantageous  point  of  loading. 

Art.  5.    Neutral  Axis 

For  pure  flexure  the  neutral  axis  will  pass  through  the 
center  of  gravity  of  the  section,  assuming  some  direction,  as 
for  example  nn  (Fig.  i),  which  direction  will  depend  upon 
the  plane  of  loading  and  upon  the  section,  but  the  neutral 
axis  will  not  necessarily  be  perpendicular  to  the  plane  of 
loading. 

In  Art.  2,  the  relation  of  a  to  0  (see  Eq.  n),  was 
obtained  in  the  derivation  of  the  expression  for  the  section 
modulus. 


10  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

This  equation  and  two  of  its  equivalent  forms  are: 

IX—J  tan  6  f    x 

tana  =  - — —  — -, (20) 

J-Iy  tan  0 

ixcot  e-j  f   N 

tana  =  - —       — , (21) 

/cot  e-iy 


tan«  = 


y 

Ix  cos  6—J  sin  0 
/cos  B—Iy  sin  0* 


For  a  combination  of  flexure  and  direct  loading  the 
neutral  axis  will  not  pass  through  the  center  of  gravity  of  the 
section,  but  it  will  be  parallel  to  the  position  which  it  would 
have  for  pure  flexure,  and  the  distance  between  the  two  par- 
allel lines  will  be, 

_F  (y  cos  a— x  sin  a)  ,    ^ 

A  / 

in  which      v  =  distance  from  the  neutral  axis  to  the  center 

of  gravity; 

F  =  total  stress  acting  normal  to  the  section; 
A  =area  of  the  section; 

/"=unit  stress  at  the  extreme  fiber  due  to  flexure; 
x,  y  =  coordinates  of  the  extreme  fiber; 
a  =  angle  which  the  neutral  axis  makes  with  X-X. 

By  the  use  of  Eqs.  (20)  to  (23)  the  position  and  the  direc- 
tion of  the  neutral  axis  can  be  determined,  but  by  the  use  of 
the  section  modulus  polygon  these  computations  can  usually 
be  avoided  for  practical  problems.  For  example,  consider  .an 
I-beam,  as  indicated  in  Fig.  6.  For  a  loading  in  the  plane 
OP,  the  section  modulus  is  represented  by  the  line  from  0  to 
the  vertex  ab.  Had  the  plane  of  loading  intersected  the  sides 
a  and  c  of  the  polygon,  the  critical  stresses  would  have  been 
at  A  and  C  of  the  beam,  and  had  the  plane  of  loading  inter- 


THEORY  AND  DISCUSSION 


11 


ab 


sected  the  sides  b  and  d  of  the  polygon,  the  points  of  critical 
stress  would  have  been  B  and  Z>,  but  since  the  plane  of  loading 
passes  through  the  vertex  ab  the  bending  stresses  at  A  and  B 
are  equal,  or  the  neutral  axis  must  be  parallel  to  the  line 
joining  A  and  B.  Also,  if  the 
plane  of  loading,  for  pure  flexure, 
is  parallel  to  a  side  of  the  sec- 
tion modulus  polygon,  the  neu- 
tral axis  will  intersect  the  corre- 
sponding vertex  of  the  section. 

As  another  example,  let  the 
plane  of  loading  be  taken  as 
OP'  (Fig.  6).  The  lineOtf  now  da( 
represents  the  section  modulus 
for  the  point  B,  and  by  pro- 
longing the  side  c  to  c',  Ocf  is 
obtained  which  is  the  section 
modulus  for  the  point  C  of  the 
beam.  The  bending  stress  at 
B  is  to  the  bending  stress  at 
C  as  Oc'  is  to  Ob'.  Since  the 

stress  is  proportional  to  the  distance  from  the  neutral  axis, 
the  distance  from  C  along  the  prolongation  of  the  line  join- 
ing B  and  C  to  the  point  N  at  which  the  neutral  axis  for 
pure  flexure  is  intersected,  will  be, 


CN  = 


BC 


(24) 


ov 


—  I 


To  observe  the  conditions  which  obtain  for  the  angle, 
consider  the  section  and  polygon  shown  in  Fig.  4.  It  will  be 
seen  that  when  the  plane  of  loading  coincides  with  OG  the 
section  modulus  for  the  side  DE  of  the  angle  is  represented 
by  the  line  from  0  to  the  vertex  de  of  the  polygon.  Since  the 


12  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

plane  of  loading  (OG)  intersects  the  vertex  de,  the  neutral  axis 
must  be  parallel  to  DE  of  the  section,  as  has  been  explained 
for  the  case  of  the  I-beam.  However,  while  for  the  I-beam 
the  neutral  axis  happened  to  be  perpendicular  to  the  plane 
of  loading,  in  the  case  of  the  angle  such  an  assumption  is  not 
even  approximately  correct.  A  study  of  the  angle  and  its 
polygon  will  show  that  if  the  plane  of  loading  coincides  with 
OF,  the  stresses  at  D  and  at  E,  for  pure  flexure,  will  be  of 
opposite  sign,  since  the  lines  d  and  e  of  the  polygon  cut  the 
plane  of  loading  on  opposite  sides  of  O;  or,  for  pure  flexure 
with  OF  as  the  plane  of  loading  the  neutral  axis  will  inter- 
sect the  side  DE  of  the  angle  within  the  limits  of  the  section. 
As  the  plane  of  loading  shifts  from  OG  to  OF  the  change  of 
sign  for  the  stress  at  E  will  occur  when  the  neutral  axis 
passes  through  E,  and  the  neutral  axis  will  intersect  E  when 
the  plane  of  loading  is  parallel  to  the  side  e  of  the  polygon. 

Art.  6.    Plane  of  Loading 

For  those  cases  in  which  the  member  is  restrained  from 
bending  in  some  direction,  the  determination  of  the  equiva- 
lent plane  of  loading  may  require  some  study.  If  the  char- 
acter of  the  restraint  determines  the  direction  in  which 
bending  must  occur,  then  the  position  of  the  equivalent  plane 
of  loading  can  be  determined  by  noting  the  direction  of  the 
neutral  axis.  Since  the  deformation  is  proportional  to  the 
distance  from  the  neutral  axis,  the  neutral  axis  must  be 
parallel  to  the  line  joining  any  two  points  whose  deformations 
are  equal  in  amount  and  of  the  same  sign.  In  such  a  case 
tan  a  becomes  a  known  quantity,  and  the  position  of  the 
equivalent  plane  of  loading  may  be  computed  by  the  use  of 
Eqs.  (20)  to  (22).  From  Eq.  (20)  the  corresponding  equation 
for  6  becomes 

„    Ix— /tana  /     N 

tane= (2S) 


THEORY  AND  DISCUSSION  13 

For  practical  purposes  the  section  modulus  polygon  may 
be  used  to  determine  the  equivalent  plane  of  loading,  avoiding 
the  necessity  for  using  Eq.  (25).  For  illustration,  consider 
again  the  angle  and  its  section  modulus  polygon,  shown  in 
Fig.  4.  When  the  neutral  axis  is  parallel  to  DE  of  the  angle, 
the  section  modulus  is  represented  by  the  line  from  0  to  the 
vertex  de,  or  if  the  angle  is  riveted  to  a  connection  plate  along 
the  face  DE,  the  angle  being  restrained  from  rotation  parallel 
to  the  plate,  and  if  the  point  of  application  be  taken  at  the 
face  of  the  connection  plate,  then  the  equivalent  point  of 
application  of  the  load  is  G.  If  the  gauge  line,  and,  there- 
fore, the  actual  point  of  attachment,  is  at  H,  the  restraining 
couple  which  is  called  into  action  will  have  a  moment  arm 
equivalent  to  GH.  It  is  sometimes  thought  that  complete 
restraint  parallel  to  the  connection  plate  implies  the  exist- 
ence of  a  couple  having  a  moment  arm  equivalent  to  MH, 
but  this  is  not  the  case;  complete  restraint  parallel  to  the 
connection  plate  will  move  the  equivalent  point  of  applica- 
tion from  either  M  or  H  to  G,  or,  rather,  the  equivalent 
plane  of  loading  to  the  plane  OG.  When  the  neutral  axis 
is  parallel  to  DE  it  will  also  be  parallel  to  BC,  and  hence,  the 
plane  of  loading  which  includes  O  and  the  vertex  de  must  also 
pass  through  the  vertex  be. 

Art.  7.    Combined  Stresses 

In  determining  the  critical  stresses  for  a  combination 
of  direct  tension  or  compression  with  flexure,  the  usual 
methods  may  be  used,  substituting  the  correct  value  of 
the  section  modulus  for  the  more  commonly  used  special 
value. 

For  a  member  which  is  short  enough  to  be  considered  as  a 
prism, 

>- '«-» 


14  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

in  which    /=unit  stress  due  to  both  flexure  and  direct  load; 
F  =  total  load  or  force  normal  to  the  section; 
A  =area  of  the  section; 
M  =  bending  moment  at  the  section; 
5  =  section  modulus. 

If  the  bending  moment  (M)  in  the  last  equation  is  due  to 
an  eccentricity  (e)  of  the  force  (F),  then, 


(27) 


For  a  slender  member,  subject  to  both  direct  loading  and 
bending,  the  following  formula  is  applicable,  and  will  be  found 
convenient  on  account  of  its  wide  range  of  application: 


<-> 


KEz 


in  which  M  =  bending  moment  neglecting  the  increment  of 
the  moment  which  is  due  to  the  deflection  of 
the  member  itself; 

L  =  length  of  the  member  (in  inches  if  A  is  in  square 
inches) ; 

E  =  coefficient  of  elasticity  of  the  material; 
z= distance  from  the  extreme  fiber  to  a  line  through 
the  center  of  gravity  of  the  section,  parallel 
to  or  coinciding  with  the  neutral  axis; 

K  —  a  constant,  the  value  of  which  is  9.6  for  a  simple 
beam  uniformly  loaded  and  12  for  a  simple 
beam  with  a  load  at  the  center,  and  for  which 
the  value  10  is  suggested  for  general  purposes 
of  design, 

Other  quantities  in  the  last  equation  are  as  denned  under 
Eq.  (26). 


THEORY  AND  DISCUSSION  15 

In  Eq.  (28),  the  negative  sign  is  used  in  the  denominator 
for  compression  and  the  positive  sign  for  tension.. 

The  stress  computed  by  Eq.  (28)  will  be  the  stress  at  the 
point  of  maximum  deflection,  which  for  a  member  in  com- 
pression will  evidently  be  the  critical  stress,  but  fo"r  a  member 
in  tension  the  deflection  reduces  the  eccentricity  at  the  mid- 
length,  and  for  this  case  the  critical  stress  will  usually  be 
at  an  end  section.  For  the  stress  at  the  end  section,  the  com- 
putation may,  in  general,  be  made  by  use  of  Eq.  (26),  but 
for  angles  the  increase  in  stress  on  account  of  the  outstanding 
legs  may  need  special  consideration, 

Art.  8.    Flexure  for  Angles  in  Pairs 

The  case  of  two  angles  of  the  same  section,  riveted  together 
at  frequent  intervals,  and  subjected  to  bending  in  a  plane 


00          0| 


FIG.  7. 

parallel  to  the  connected  legs,  is  indicated  in  Fig.  7.  For 
such  a  condition  the  restraint  which  each  angle  exerts  upon 
the  other  may  be  assumed  to  produce  a  deflection  in  the  plane 
of  bending  and  the  neutral  axis  may  be  assumed  to  be  per- 
pendicular to  the  connected  legs.  The  loading  for  each  angle 
will  then  be  equivalent  to  some  loading  in  a  plane  passing 
through  0,  the  center  of  area  of  the  section,  and  through  the 
vertices  be  and  de  of  the  section  modulus  polygon,  which  is 
indicated  for  each  angle  in  Fig.  8.  Let  m  denote  the  bending 
moment  for  one  angle,. which  if  acting  in  the  plane  of  0,  be, 
and  de  would  produce  the  same  deformations  as  those  which 
result  from  the  actual  loading  for  the  pair  of  angles.  The 


16 


STRESSES  IN  STRUCTURAL  STEEL  ANGLES 


actual  moment  must  be  equal  to  the  resultant  of    the  two 
moments  m,  or 

M  =  2m  sin  6,       .     .     .     .        (29) 

in  which  M  =  actual  bending  moment  for  the  pair  of  angles; 
m  =  equivalent  bending  moment  for  one  angle ; 
6  =  angle  which  the  plane  for  m  makes  with  the 
.neutral  axis  for  the  pair  of  angles, 


de 


FIG.  8. 


The  bending  stress  for  each  angle  is  equal  to  m  divided  by 
the  section  modulus  in  its  plane,  or 


M 


2S  sin  0' 


(30) 


in  which    /  =  unit  stress  at  the  extreme  fiber; 

s  =  section  modulus  for  the  plane  of  m. 


In  Eq.  (30),  the  value  of  s  sin  6  may  be  considered  as  the 
equivalent  section  modulus  for  one  angle  of  the  pair  for  half 
of  the  total  vertical  load.  For  the  case  illustrated,  this  value 
is  equal  to  the  ^-coordinate  of  be,  for  the  edge  BC  of  the  angle, 
or  to  the  ^-coordinate  of  de  for  the  edge  DE  of  the  angle. 


THEORY  AND  DISCUSSION 


17 


Thus,  for  angles  in  pairs,  the  equivalent  section  modulus  for 
one  angle  can  be  obtained  directly  from  a  table  of  coordinates 
of  section  modulus  polygons.  However,  even  this  is  unnec- 
essary if  a  table  of  properties  of  sections  is  at  hand,  for  the 
values  of  the  coordinates  in  question  are  each  obtained  by 
dividing  the  moment  of  inertia  by  the  distance  from  the 
neutral  axis  to  the  extreme  fiber.  In  other  words,  the  stress 
for  the  pair  of  angles,  for  bending  in  the  plane  of  symmetry, 
may  be  computed  as  for  a  single  symmetrical  section,  provided 
the  two  angles  are  firmly  attached  to  each  other.  For  the 
case  shown  in  Fig.  7,  independent  action  upon  the  part  of 
each  angle  would  tend  to  crowd  the  angles  toward  each  other 
at  the  midspan. 


i 


Art.  9.    Transfer  of  Stress  by  Shear  to  an  Outstanding  Leg 

In  a  member  consisting  of  one 
or  more  angles,  the  critical  stress 
is  likely  to  occur  at  a  section  near 
the  connection  of  the  member  to 
one  of  its  connection  planes,  par- 
ticularly in  the  case  of  tension 
members.  At  such  a  section  there 
may  be  considerable  variation  in 
the  distribution  of  the  stress,  due 
to  a  difference  in  the  deformations 
of  the  connected  and  the  out- 
standing legs  of  each  angle.  In 
order  to  make  an  approximate 
determination  of  the  stress  dis- 
tribution, due  to  the  transfer  of 
stress  to  the  outstanding  leg  by 
shear,  consider  the  case  shown  in 
Fig.  9,  in  which  there  are  two 
angles  connected  to  opposite  sides 
of  a  connection  plate.  In  this  case  the  stress  at  a  section 


6' 

r''        i 

6 

' 

a      ' 

C 

) 

c 

) 

c 

} 

d' 

d 

L 

1 

1 

ilium 


FIG.  9. 


18  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

taken  at  the  middle  of  the  length  of  the  member  will  be 
assumed  *  to  be  uniformly  distributed,  but  at  the  end  of  the 
member  the  point  a  is  subject  to  slightly  less  deformation  than 
the  point  b.  If  it  be  assumed  that  the  stress  increases  at  a 
uniform  rate  from  zero  at  point  a  to  the  average  unit  stress  at 
the  center  of  the  length  of  the  member,  then  at  any  section 
distant  x  from  the  point  a  the  unit  stress  at  the  extreme  fiber 
of  the  outstanding  leg  will  be 

,        2X    F 


in  which  /'  =  unit  stress  at  the  extreme  fiber  of  the  outstand- 

ing leg; 
x  =  distance  from  the  end  of  the  angle  to  the  sec- 

tion at  which  f  is  taken; 
L  =  length  of  the  angle; 
F  =  total  stress  or  force  acting  lengthwise  of  the 

angle; 
A  =  area  of  cross-section  of  the  angle. 

Also,  let  /  be  the  unit  stress  in  the  connected  leg  of  the 
angle  at  the  section  at  which/7  is  taken.  Then,  for  equilibrium 
at  the  section  x  distant  from  the  end  of  the  angle, 

.....     (32) 


in  which  A  i  =  area  of  the  outstanding  leg; 
A  2  =  area  of  the  connected  leg. 

Or,  substituting  the  value  of/  from  Eq.  (31)  in  Eq.  (32), 
and  solving  for/, 


/=2 


x 

I~ 


A 


(33) 


*  The  reasonableness  of  the  assumptions  of  this  article  will  be  better  under- 
stood by  a  study  of  the  results  of  tests  by  Prof.  Cyril  Batho,  reported  in  an 
article  on  "  The  Effect  of  End  Connections  on  the  Distribution  of  Stress  in 
Certain  Tension  Members,"  Journal  of  the  Franklin  Institute,  August,  1915. 


THEORY  AND  DISCUSSION  19 

Art.  10.    Efficiency  of  End  Connections 

The  efficiency  of  a  member  may  be  taken  as  the  relation  of 
the  average  to  the  maximum  stress,  or  if,  in  Eq.  (33),  x  is  taken 
so  that  /  will  be  the  maximum  stress,  the  equation  for  the 
efficiency  for  the  transfer  of  stress  by  shear  to  the  outstanding 
leg  will  be, 

_,      iA+A2 


in  which  EI  is  the  efficiency,  and  the  other  quantities  are 
as  denned  under  Eqs.  (31)  and  (32). 

In  place  of  the  areas  A\  and  A  2  the  widths  of  the  legs  of 
the  angle  may  be  used.  Using  il\  for  AI  and  tl2  for  A2>  the 
equations  corresponding  to  Eqs.  (32),  (33),  and  (34)  become 


•    •    .    (35) 

SI  2 

or 

.  .  (36) 

^1  L      H  T^2      J 

and 


For  use  in  designing,  the  critical  section  will  usually  be 
taken  through  the  connection  rivet  or  rivets  most  distant 
from  the  end  of  the  angle,  in  which  case  the  net  areas  of  the 
legs  will  be  used  for  A\  and  A2,  or  in  Eqs.  (35),  (36)  and  (37), 
the  values  of  the  widths  of  legs  must  be  corrected  to  their  net 
values.  For  such  a  case  the  equations  become 

n2d),      .     (38) 


20  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

or 


and 

Ei=- 


in  which    /= maximum  unit  stress  at  the  critical  section; 

EI  =  efficiency  of  net  section  relative  to  gross  section ; 

/i  =  width  of  the  outstanding  leg;* 

/2  =  width  of  the  connected  leg;* 

ni=  number  of  rivet  holes  in  the  outstanding  leg 
which  are  cut  by  the  critical  section; 

H2  =  number  of  rivet  holes  in  the  connected  leg  which 
are  cut  by  the  critical  section; 

d  =  diameter  of  each  rivet  hole ; 

x  =  distance  from  the  end  of  the  angle  to  the  crit- 
ical section; 

L  =  total  length  of  angle. 

For  a  connection  with  lug  angles,  the  length  of  the  lug 
may  be  added  to  both  x  and  — ,  in  the  equations  just  given,  but 

2 

this  must  be  considered  as  a  rather  crude  approximation. 

Since  the  equations  of  articles  9  and  10  are  developed  upon 
the  assumption  that  the  angle  is  restrained  from  bending, 
they  are  applicable  in  particular  to  pairs  of  angles  riveted 
to  opposite  sides  of  a  connection  plate,  but,  as  will  be  shown 
later,  they  may  also  be  used  in  the  design  of  single  angles. 
Structural  specifications  are  inclined  to  be  both  indefinite 
and  unsatisfactory  with  reference  to  the  requirements  for 
connections.  If  any  clause  relating  to  this  matter  is  included, 

*  For  exact  work  reduce  the  outside  dimension  of  the  leg  by  half  the  thick- 
ness of  the  angle. 


THEORY  AND  DISCUSSION  21 

it  usually  stipulates  that  each  angle  must  be  connected  by 
both  legs  or  that  only  one  leg  shall  be  considered  as  effective 
area.  The  results  of  tests  indicate  that  the  allowance  of  only 
one  leg  as  effective  area  is  too  rigid  a  requirement,  even 
though  but  one  leg  of  the  angle  is  connected;  while  on  the 
other  hand  the  use  of  a  lug  angle  to  connect  the  outstanding 
leg  results  in  too  liberal  an  allowance  for  the  capacity  of  the 
angle.  Eq.  (40)  furnishes  a  ready  means  for  estimating  the 
efficiency  of  angles  used  in  pairs,  and  as  already  mentioned 
may  be  used  for  connections  with  or  without  lug  angles.  For 
an  angle  without  lugs,  n\  will  become  zero.  For  this  case  the 
author  has  computed  the  efficiency  of  those  angles  which  are 
most  used,  the  values  for  which  are  given  in  Tables  IV  and  V, 
at  the  back  of  the  book.  In  the  same  tables,  there  is  given 
for  each  angle,  the  area  which  if  entirely  effective  at  the  max- 
imum unit  stress  would  be  equal  in  strength  to  the  connected 
angle  for  the  computed  efficiency.  Also,  the  total  allowable 
stress  for  a  maximum  unit  stress  of  16,000  pounds  per  square 
inch  at  the  critical  section  is  tabulated. 

In  the  case  of  a  single  angle  attached  to  a  connection  plate, 
there  may  be  nearly  complete  restraint  parallel  to  the  plate 
with  but  little  restraint  perpendicular  to  the  plate.  In  this 
case  bending  and  shear  each  have  an  effect,  but  within  the 
limits  of  working  stresses  the  effect  of  the  longitudinal  shear 
may  usually  be  neglected.  In  the  development  of  Eqs.  (31) 
to  (40),  inclusive,  it  was  assumed  that  the  stress  at  the  center 
of  the  member  was  uniformly  distributed.  While  this  con- 
dition will  be  nearly  fulfilled  for  angles  in  pairs,  a  single  angle, 
with  a  small  load,  will  be  subjected  to  bending  throughout  its 
length,  in  which  case  the  stress  at  the  extreme  edge  of  the 
outstanding  leg  will  be  less  than  the  average  stress,  or  may  be 
of  opposite  sign.  For  the  last-named  case  the  stresses  at 
the  midsection  may  be  as  indicated  in  Fig.  10,  in  which  j\ 
represents  the  unit  stress  at  the  face  of  the  connected  leg  and 
[2  the  unit  stress  at  the  extreme  fibers  of  the  outstanding  leg. 


22  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

At  some  distance  b  from  /2,  the  horizontal  shear  will  be  zero, 
and  the  stress  which  must  be  transferred  by  horizontal  shear 
from  the  connected  to  the  outstanding  leg  will  be  equal  to 
the  sum  of  the  stresses  between  connected  leg  and  the  section 
Ja  of  zero  horizontal  shear.  Furthermore,  if  it 

be  assumed  that  the  angle  is  restrained  from 
bending  until  the  horizontal  shear  necessary 
to  transfer  stress  to  the  outstanding  leg  has 
occurred,  then  at  the  critical  section  the 
moment  arm  tending  to  produce  bending  will 
FIG.  10.  ke  the  distance  from  the  centroid  of  the 

stresses  at  the  section  to  the  line  of  action  of 
the  applied  force  in  the  connection  plate.  But,  on  account 
of  the  variation  of  the  stresses  across  the  section,  due  to 
the  longitudinal  shear,  the  centroid  of  the  stresses  will  lie 
between  the  center  of  area  of  the  section  and  line  of  action 
of  the  force  in  the  connection  plate.  In  other  words  the 
neglect  of  the  horizontal  shear  for  this  condition  will  be 
partially  counterbalanced  by  the  reduction  in  the  moment 
arm  due  to  the  shear,  and  designs  may  be  made  on  the  basis 
of  the  bending  and  direct  stresses.  In  this  case  for  one 
angle  in  tension,  the  stress  may  be  computed  by  Eq.  (27), 
or  the  efficiency  for  direct  tension  and  flexure  will  be 


in  which  Ez  =  efficiency  of  a  single  angle  for  eccentric  ten- 
sion; 
s  =  section  modulus  of  the  critical  section  for  the 

given  loading; 
A  =area  of  section; 

e  =  eccentricity  of  the  applied  load  at  the  critical 
section. 


THEORY  AND  DISCUSSION  23 

In  using  Eq.  (41)  the  critical  section  will  be  at  some 
distance  from  the  extreme  end  of  the  angle.  Hence,  the  value 
of  e  to  be  used  will  be  less  than  that  at  the  end  section,  due 
to  the  deflection  of  the  angle.  Since  this  reduction  in  e 
may  result  in  an  appreciable  increase  in  the  value  of  £2, 
its  determination  may  be  desired.  In  Fig.  n,  ec  represents 
the  resulting  eccentricity  at  the  center,  e  is  the  eccentricity 


FIG.  11. 


at  the  critical  section,  and  e\  is  the  eccentricity  at  the  end. 
The  maximum  stress  at  the  midsection  in  terms  of  ec  is 


or  in  terms  of  e\  it  is 

,     F.     Fe 


+ 


,    . 
(43) 


KEz 


in  accordance  with  Eq.  (28),  to  which  the  reader  is  referred 
for  nomenclature. 

Equating  the  values  of  /in  Eqs.  (42)  and  (43), 


^ 

KEzs 

in  which  e\  is  eccentricity  of  the  applied  load  at  the  end  sec- 
tion, and  the  other  values  are  as  stated  for  Eq.  (28). 

At  the  instant  that  the  angle  begins  to  deflect,  the 
moment  at  every  section  is  Fei  and  the  elastic  curve,  for 
this  value  of  the  moment,  would  be  a  parabola.  As  the  load 
increases,  the  deflection  at  the  midsection  approaches  zero 


24  STRESSES  IN  STRUCTURAL  STEEL  ANGLES 

as  a  limit,  and  if  for  this  condition  e  be  assumed  propor- 

/L      \n 

tional  to  ( x)  ,  the  resulting  elastic  curve  will  give  e  pro- 
XL      \n+2 
portional  to  ( x)       ,  or  the  limit  toward  which  the  curve 

(L      \°° 
tends  is  that  for  which  the  value  of  e  is  proportional  to  ( x )  . 

The  significance  of  this  may  be  appreciated  by  reference  to 

Fig.  12.  The  bending  would  all 
occur  at  the  end  of  the  unsup- 
ported length,  and  the  centroid  of 
the  stresses  throughout  this  un- 
FIG,  12.  supported  length  would  coincide 

with    the   line  of   action  of    the 

applied  forces.  In  reality  this  limit  can  not  be  reached, 
for  some  bending  occurs  throughout  the  unsupported  length, 
and  at  any  section  this  resisting  moment  must  balance  the 
moment  which  is  equal  to  the  resultant  force  F  multiplied 


FIG.  13. 

by  the  distance  from  the  centroid  of  the  stresses  to  the  line  of 
action  of  the  applied  force.  Evidently,  the  curve  of  the 
neutral  surface  for  the  unsupported  length  of  the  angle  will 
approximate  the  form  of  the  parabola,  and  at  the  end  of  the 
unsupported  length  the  maximum  bending  stresses  will  be 
developed.  For  an  angle  attached  to  a  plate  the  section  of 
the  sharp  bend  will  be  transferred  to  the  connection  plate 
and  will  occur  close  to  the  end  of  the  angle,  as  indicated  in 
Fig.  13.  Therefore,  the  curve  throughout  the  length  may 
ordinarily  be  taken  as  a  parabola,  for  which  the  value  of  e 
in  Fig.  ii  becomes 

2       ....     (45) 


THEORY  AND  DISCUSSION  25 

in  which    e  =  eccentricity  at  the  critical  section; 
ec  =  eccentricity  at  the  midsection; 
e\  —  eccentricity  at  the  end  section; 
x  =  distance  from  the  end  of  the  angle  to  the  critical 

section ; 
L  =  length  of  angle. 

It  is  evident  that  for  large  stresses  and  for  practically  all 
cases  of  ultimate  loads  the  condition  indicated  in  Fig.  12 
will  be  nearly  fulfilled  and  that  the  stresses  at  the  critical 
section  will  tend  to  become  identical  with  those  which  were 
determined  in  Art.  9.  In  this  case  the  efficiency  becomes 
identical  with  the  value  derived  for  pairs  of  angles,  as  given 
in  Eqs.-(34)  and  (40).  Of  the  two  efficiencies  which  have  been 
derived,  one  for  the  transfer  of  stress  to  the  web  and  the  other 
for  the  bending  stress,  the  lower  of  the  two  should  control  in 
the  design.  As  the  load  becomes  large  the  value  of  e  at  the 
critical  section  becomes  small,  and  the  efficiency  for  transfer 
of  stress  becomes  dominant.  That  value  may,  therefore, 
reasonably  be  used  as  a  basis  for  design,  for  tension  members. 

It  may  be  of  interest  to  compare  some  values  obtained  by 
experiment  with  the  values  for  efficiency  computed  by  Eq. 
(40).  Table  A  contains  a  summary  of  the  average  values  for 
tests  made  by  Prof.  F.  P.  McKibben  and  reported  in  Engi- 
neering News,  Vol.  56,  page  14,  and  Vol.  58,  page  190. 

In  studying  the  results  for  Table  A,  it  should  be  remem- 
bered that  Eq.  (40)  is  developed  for  a  straight  line  relation  of 
stresses  at  the  critical  section,  whereas  at  the  ultimate  load 
the  stress  is  not  proportional  to  the  deformation  and  that  in 
this  case  it  will  cause  the  computed  efficiency  to  be  less  than 
that  observed  for  the  tests.  On  the  other  hand,  influences 
which  produce  bending  parallel  to  the  outstanding  leg  of  the 
connection  tend  to  make  the  efficiency  obtained  by  experi- 
ment lower  than  the  efficiency  computed  by  Eq.  (40),  which 
neglects  this  bending.  On  the  whole  the  results  obtained 


26 


STRESSES  IN  STRUCTURAL  STEEL  ANGLES 


TABLE  A 

COMPARISON  OF  COMPUTED  AND  OBSERVED  EFFICIENCIES  OF  CONNECTIONS 
FOR  SPECIMENS  TESTED  BY  PROF.  .McKiBBEN 


MAIN  ANGLES.* 

LUG  ANGLES. 

Holes 

EFFICIENCY 

IN  PER  CENT 

1 

o 

m 

3 

^ 

' 

^JH 

fe 

(M 

d 

0 

O 

Remarks. 

fl 

s 

«J 

Jj 

§ 

h 

h 

L'gth 

| 

Length. 

II 

X 

"3 

0" 

H 

S   <u 

1 

fe 

In 

In. 

Li 

? 

h 

22 

L2 

Ft. 

Ft. 

m 

ns 

fi 

ffl 

A! 

1 

3 

3* 

5'-4" 

o 

5.33 

1.20 

0 

69.4 

68.0 

,A2 

1 

3 

3f 

5'-4" 

J 

;• 

O'-Q^" 

6.92 

1.35 

63.5 

64.9 

A 

1 

3 

4 

5'-4" 

o 

5.33 

1.20 

( 

72.1 

64.1 

B2 

1 

3 

4 

5'-4" 

1 

J 

; 

0'-  9|" 

6.92 

1.35 

- 

61.3 

64.1 

B3 

1 

3 

4 

5'-4" 

( 

5.33 

1.20 

( 

• 

72.1 

69.1 

Ci 

1 

4 

6 

5'-4" 

( 

5.33 

1.46 

( 

t 

68.5 

70.5 

C2 

1 

4 

6 

5'-4" 

1 

. 

;: 

1'-  3" 

7.83 

2.37 

] 

2 

61.7 

72.7 

A 

o 

3 

3 

5'-4" 

o 

5.33 

1.37 

( 

• 

67.7 

64.8 

A 

2 

3 

3 

7.08 

1.50 

1 

, 

54.3 

66.4 

5'  -4" 

2 

, 

ii 

O'-IO!" 

4, 

1 

3 

3£ 

5'-4" 

1 

J 

. 

O'-IH" 

7.25 

1.52 

1 

1 

58.1 

74.2 

^4 

J 

3 

3| 

5'-4" 

1 

J 

0'-11|" 

7.25 

1.52 

1 

1 

58.1 

75.9 

^5 

1 

3 

3| 

5'-4" 

1 

,'; 

o'-iH" 

7.83 

2.37 

1 

1 

60.1 

76.4 

^    -~-  1  '    Q' 

.64 

1 

3 

4 

5'-4" 

1 

;> 

O'-iU" 

7.25 

1.52 

1 

1 

61.5 

77.4 

B5 

1 

3 

4 

5'-4" 

1 

J 

;! 

0'-11|" 

7.25 

1.52 

1 

1 

61.5 

79.0 

B6 

1 

3 

4 

5'-4" 

1 

3 

3 

O'-Hi" 

7.25 

1.52 

1 

1 

61.5 

70.3 

B7 

1 

3 

4 

5/-4// 

1 

3 

3 

O'-Hi" 

7.21 

1.50 

1 

1 

61.5 

79.1 

Bs 

1 

3 

4 

5^/7 

1 

3 

3 

O'-ll" 

7.19 

1.48 

1 

1 

61.5 

70.9 

CA 

1 

4 

6 

5'-4' 

0 

5.33 

1.51 

0 

1 

79.7 

71.6 

C4 

1 

4 

6 

5'_4" 

0 

5.33 

1.51 

0 

2 

68.7 

70.5 

C5 

1 

4 

6 

5'-4" 

1 

3 

4 

'-!" 

7.49 

2.09 

0 

2 

68.7 

69.4 

A 

2 

3 

3 

SM" 

0 

5.33 

1.37 

0 

1 

67.7 

68.9 

A 

2 

3 

3 

3'-4" 

2 

3 

3 

'-0" 

7.33 

1.83 

1 

1 

55.2 

67.4 

*  Thickness  of  metal  for  A,  A,  A,  and  Z>4  was 
specimens  f  inch, 


inch,  and  for  all  other 


THEORY  AND  DISCUSSION  27 

by  computation  appear  to  be  sufficiently  conservative  to 
warrant  the  presentation  of  the  method  for  practical  use,  until 
a  better  method  shall  be  proposed. 

For  a  single  angle  in  compression,  the  critical  section  for 
bending  will  be  at  the  mid-length,  the  maximum  stress  may  be 
computed  by  Eq.  (28),  and  the  efficiency  for  direct  compres- 
sion and  flexure  will  be 


KEz 

in  which  E$  =  efficiency  of  a  single  angle  for  eccentric  com- 

pression; 
s  =  section  modulus  of  the  critical  section  for  the 

given  loading; 
A  =  area  of  section  ; 
e=  moment  arm  at  the  end  section  of  the  couple 

producing  flexure; 
F  =  total  applied  force; 
/  =  length  of  member; 

K  =  a  constant,  for  which  the  value  10  may  be  used; 
E=  modulus  of  elasticity  of  steel,  usually  taken  as 

either  29,0x30,000  or  30,000,000; 
2  =  distance  from  the  extreme  fiber  to  a  line  through 
the  center  of  area  of  the  section  and  parallel 
to  the  neutral  axis. 

The  efficiency  of  the  connection  may  be  taken  identical 
with  the  values  given  for  tension,  and  if  this  is  lower  than  the 
value  obtained  from  Eq.  (42),  it  should  govern  the  design. 


TABLES 


TABLE  I 

ELEMENTS  OF  ANGLES  WITH  EQUAL  LEGS  * 

.i  F 

8    !  /  =  moment  of  inertia 

jL  «/= product  of  inertia 

^L~_£^~_~jx2  r- radius  of  gyration  « 


Weight 

Area 

Axis  1- 

• 

1   and  A 

xis  2-2. 

Axis 
3-3. 

Size. 

Foot. 

of 
Section. 

I 

r 

I 
l-x 

z 

j 

r  min. 

Inches. 

Pounds. 

In.2 

In.* 

In. 

In.s 

In. 

In.* 

In. 

8X8XU 

56.9 

16.73 

98.0 

2.42 

17.5 

2.41 

57.3 

1.55 

8X8X1;^ 

54.0 

15.87 

93.5 

2.43 

16.7 

2.39 

54.7 

1.56 

8X8X1 

51.0 

15.00 

89.0 

2.44 

15.8 

2.37 

52.1 

1.56 

8X8XM 

48.1 

14.12 

84.3 

2.44 

14.9 

2.34 

49.6 

.56 

8X8X1 

45.0 

13.23 

79.6 

2.45 

14.0 

2.32 

47.1 

.56 

8X8XH 

42.0 

12.34 

74.7 

2.46 

13.1 

2.30 

44.2 

.57 

8X8X1 

38.9 

11.44 

69.7 

2.47 

12.2 

2.28 

41.2 

.57 

8X8XH 

35.8 

10.53 

64.6 

2.48 

11.2 

2.25 

38.3 

.58 

8X8X1 

32.7 

9.61 

59.4 

2.49 

10.3 

2.23 

35.4 

.58 

8X8XA 

29.6 

8.68 

54.1 

2.50 

9.3 

2.21 

32.1 

.58 

8X8X£ 

26.4 

7.75 

48.6 

2.51 

8.4 

2.19 

28.9 

.58 

6X6X1 

37.4 

11.00 

35.5 

1.80 

8.6 

1.86 

20.6 

.16 

6X6XH 

35.3 

10.37 

33.7 

1.80 

8.1 

1.84 

19.6 

.16 

6X6X1 

33.1 

9.73 

31.9 

1.81 

7.6 

1.82 

18.6 

.17 

6X6XH 

31.0 

9.09 

3Q.  1 

1.82 

7.2 

1.80 

17.5 

.17 

6X6X1 

28.7 

8.44 

28.2 

1.83 

6.7 

1.78 

16.4 

.17 

6X6XH 

26.5 

7.78 

26.2 

1.83 

6.2 

1.75 

15.4 

.17 

6X6X1 

24.2 

7.11 

24.2 

1.84 

5.7 

1.73 

14.3 

.17 

6X6X& 

21.9 

6.43 

22.1 

1.85 

5.1 

1.71 

13.1 

.18 

6X6X£ 

19.6 

5.75 

19.9 

1.86 

4.6 

1.68 

11.9 

.18 

6X6X3^ 

17.2 

5.06 

17.7 

1.87 

4.1 

1.66 

10.6 

.19 

6X6X1 

14.9 

4.36 

15.4 

1.88 

3.5 

1.64 

9.2 

.19 

*  The  values  of  J  were  computed  by  the  author.  The  remaining  values  in  this 
table  are  from  the  Carnegie  Pocket  Companion,  and  are  used  with  permission  of  the 
Carnegie  Steel  Company. 

31 


32 


TABLES 


TABLE  I— Continued 
ELEMENTS  OF  ANGLES  WITH  EQUAL  LEGS 


/  =  moment  of  inertia 
J  =  product  of  inertia 
r— radius  of  gyration 


F 
Ib 


Weight 

Area 

Axis  1- 

1    AND    A 

xi  s  2-2. 

Axis 
3-3. 

Size. 

Foot. 

of 
Section. 

7 

r 

/ 
l-x 

X 

j 

r  min. 

Inches. 

Pounds. 

In.* 

In." 

In. 

In.  3 

In. 

In." 

In. 

5X5X1 

30.6 

9.00 

19.6 

.48 

5.8 

1.61 

11.1 

0.96 

5X5XH 

28.9 

8.50 

18.7 

.48 

5.5 

1.59 

10.6 

0.96 

5X5X1 

27.2 

7.98 

17.8 

.49 

5.2 

1.57 

10.1 

0.96 

5X5XH 

25.4 

7.47 

16.8 

.50 

4.9 

1.55 

9.6 

0.97 

5X5X1 

23.6 

6.94 

15.7 

.50 

4.5 

1.52 

9.2 

0.97 

5X5XH 

21.8 

6.40 

14.7 

.51 

4.2 

1.50 

8.6 

0.97 

5X5X1 

20.0 

5.86 

13.6 

1.52 

3.9 

1.48 

8.0 

0.97 

5X5XA 

18.1 

5.31 

12.4 

1.53 

3.5 

1.46 

7.4 

0.98 

5X5X1 

16.2 

4.75 

11.3 

1.54 

3.2 

1.43 

6.7 

0.98 

5X5X& 

14.3 

4.18 

10.0 

1.55 

2.8 

1.41 

6.1 

0.98 

5X5X1 

12.3 

3.61 

8.7 

1.56 

2.4 

1.39 

5.4 

0.99 

4X4XM 

19.9 

5.84 

8.1 

1.18 

3.0 

1.29 

4.6 

0.77 

4X4X1 

18.5 

5.44 

7.7 

1.19 

2.8 

1.27 

4.4 

0.77 

4X4XH 

17.1 

5.03 

7.2 

1.19 

2.6 

1.25 

4.1 

0.77 

4X4X1 

15.7 

4.61 

6.7 

1.20 

2.4 

1.23 

3.8 

0.77 

4X4X^ 

14.3 

4.18 

6.1 

1.21 

2.2 

1.21 

3.5 

0.78 

4X4X1 

12.8 

3.75 

5.6 

1.22 

2.0 

1.18 

3.2 

0.78 

4X4Xtfe 

11.3 

3.31 

5.0 

1.23 

1.8 

1.16 

2.9 

0.78 

4X4X1 

9.8 

2.86 

4.4 

1.23 

1.5 

1.14 

2.6 

0.79 

4X4X& 

8.2 

2.40 

3.7 

1.24 

1.3 

1.12 

2.2 

0.79 

4X4Xi 

6.6 

1.94 

3.0 

1.25 

1.0 

1.09 

1.8 

0.79 

3£X3£XH 

17.1 

5.03 

5.3 

1.02 

2.3 

1.17 

2.9 

0.67 

3*X3*Xf 

16.0 

4.69 

5.0 

.03 

2.1 

1.15 

2.8 

0.67 

3£X3^XH 

14.8 

4.34 

4.7 

.04 

2.0 

1.12 

2.6 

0.67 

3^X3^X1 

13.6 

3.98 

4.3 

.04 

1.8 

1.10 

2.5 

0.68 

3£X3£Xfk 

12.4 

3.62 

4.0 

.05 

1.6 

1.08 

2.3 

0.68 

3|X3£X* 

11.1 

3.25 

3.6 

.06 

1.5 

1.06 

2.1 

0.68 

TABLES 


33 


TABLE  I— Concluded 
ELEMENTS  OF  ANGLES  WITH  EQUAL  LEGS 


7  =  moment  of  inertia 
J  =  product  of  inertia 
r= radius  of  gyration 


F 
L, 


Size. 

Weight 
Foot. 

Area 
of 
Section. 

AXIS    1-1    AND  AXIS   2-2. 

Axis 
3-3. 

/ 

r 

J 

l-x 

X 

J 

r  min. 

Inches. 

Lbs. 

In.2 

In.< 

In. 

In.' 

In. 

In.* 

In. 

3|X3£Xf* 

9.8 

2.87 

3.3 

1.07 

1.3 

1.04 

1.9 

0.68 

3iX3|X| 

8.5 

2.48 

2.9 

1.07 

1.2 

1.01 

1.7 

0.69 

3fX3^XA 

7.2 

2.09 

2.5 

1.08 

0.98 

0.99 

1.5 

0.69 

3^X3£Xi 

5.8 

1.69 

2.0 

1.09 

0.79 

0.97 

1.2 

0.69 

3X3X1 

11.5 

3.36 

2.6 

0.88 

1.3 

0.98 

1.5 

0.57 

3X3X& 

10.4 

3.06 

2.4 

0.89 

1.2 

0.95 

1.4 

0.58 

3X3X1 

9.4 

2.75 

2.2 

0.90 

1.1 

0.93 

1.3 

0.58 

3X3X& 

8.3 

2.43 

2.0 

0.91 

0.95 

0.91 

1.1 

0.58 

3X3Xf 

7.2 

2.11 

1.8 

0.91 

0.83 

0.89 

1.0 

0.58 

3X3XA 

6.1 

1.78 

1.5 

0.92 

0.71 

0.87 

0.87 

0.59 

3X3X£ 

4.9 

1.44 

1.2 

0.93 

0.58 

0.84 

0.75 

0.59 

2^X2^X1 

7.7 

2.25 

1,2 

0.74 

0.73 

0.81 

0.68 

0.47 

2|X2^X& 

6.8 

2.00 

1.1 

0.75 

0.65 

0.78 

0.62 

0.48 

2^X2^X1 

5.9 

1.73 

0.98 

0.75 

0.57 

0.76 

0.57 

0.48 

2|X2£XA 

5.0 

1.47 

0.85 

0.76 

0.48 

0.74 

0.49 

0.49 

2^X21X1 

4.1 

1.19 

0.70 

0.77 

0.39 

0.72 

0.41 

0.49 

2^X2£XA 

3.07 

0.90 

0.55 

0.78 

0.30 

0.69 

0.32 

0.49 

2^X2|Xi 

2.08 

0.61 

0.38 

0.79 

0.20 

0.67 

0.23 

0.50 

2X2X;& 

5.3 

1.56 

0.54 

0.59 

0.40 

0.66 

0.30 

0.39 

2X2X1 

4.7 

1.36 

0.48 

9.59 

0.35 

0.64 

0.27 

0.39 

2X2XA 

3.92 

1.15 

0.42 

0.60 

0.30 

0.61 

0.24 

0.39 

2X2X1 

3.19 

0.94 

0.35 

0.61 

0.25 

0.59 

0.21 

0.39 

2X2X& 

2.44 

0.71 

0.28 

0.62 

0.19 

0.57 

0.16 

0.40 

2X2Xi 

1.65 

0.48 

0.19 

0.63 

0.13 

0.55 

0.11 

0.40 

34 


TABLES 


B     g 


e 


J 


1 1 1 

fl     S     r 


I   « 

O       M       TO 

1    ft  "g 


n 

h 


T; 


/ 


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TABLES 


35 


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36 


TABLES 


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a  ^  o 

2  3  3 

S  ^3  .2 

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TABLES 


37 


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OOOOGOOOOOOOC5O5OSCiC5O5  iO 


C^l   00   T*|   Oi   CO   t^-   I-H    rf  i    CO    00    Oi    C5  Tj<    »O   «O   CO   CO   O   »O   CO 

cst>io^cO'-HOoocOT^oio        oiOTjHcofNi-Idci 


i—  i»OOO<N«OOOO 


1— I  ^Hpl  H«  iH(<H  «Hl  MM 

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r^|«    i-»IW    r^|M    H«    H«    HW    HW    HW    HO*    H«    H^    HW 

CO  CO  CO  £O  CO  CO  CO  CO  CO  CO  CO  CO  ^^  ^^  ^^  ^^  T^  '^  Tf  T^  ^< 

XXXXXXXXXXXX  XXXXXXXXX 

CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  *O  *O  *O  *O  ^O  ^O  ^O  ^O  &O 


38 


TABLES 


f 


F 


•J  I 


§        §        « 

a  -§  .3 

I  §,1 
ii   ii   ii 

K,     ^      K 


B| 


c3        o 

2*0 '-£ 


oooooooooo 


(M   (M   <N   (N 


OOOO 


XXXXXXXXXX 

coeococococococococo 
XXXXXXXXXX 


TABLES  39 


TjH-tfTtH-^iO^OOiOO     -^T^^^T^iOiOCOO     <N(N<M<N<M<N(MCOCO 
OcOcOcOCOcOcOcOCO     cotOOOCOcOOcOcO     t>t>-t>-I>-I>-l>-l>l>»I>- 

dddddoOOO  OOOOOOOOO  oddddddod 


i-Hi-    co  co  co  co  < 


COT^CSQOI>-*OCOOOO      OOOiOCOi-HCiCO^fC^      i-HOJt^^fC^O 

ddddddddo    oddddddod    t-i  1-1  ,-H  i-«  ,-H  »-< 


Ci    IO 

CO    i-H    O    OC 


I>     !>O>OTfiCOi— iOGOI>      COi-H 
O  O     i-i  i-i  i-i   i-i  i-i   »-J   i-<   O  C>     ci(N 


O    O   i— i    »— iC<lCOT^TtHiO     T-H!MCOCOiOiO»OOI>      i— i    i— (C^COCOTtHiOCOb- 

OOOOOOOOGOOOOOOOOO    ooooooaqoqoooooooo    pppppcjcop 
ddddddddd    ddddddddd    ^'^^^,-1^^^^ 


QOQOGOGOl>t^I>t^O      cOOOtO»OiC>OTtiTti      OOrOCO<N<>><MC<lC^i— " 


co  co  <M  IN  c  >-  co  co  co  IM  IM  <N  <N  i-  r-  <N  <N  <N 


lOiOOt-OOOO^H-H      OCOlC50—i<NCOTt<^      C50-HC^COCO^iO<0 
IO»OIO^»OIOCOCOCO      COCOCO-^^^TfTtiTti      i— ((NOatNC^C^fMfMJN 


T— IO5GO 


XXXXXXXXX  XXXXXXXXX  XXXXXXXXX 

COCOCOCOCOCOCOCOCO  COCOCOCOCOCOCOCOCO  COCOCOCOCOCOCOCOCO 

XXXXXXXXX  XXXXXXXXX  XXXXXXXXX 
f 


• 


40 


TABLES 


r 


•S    S    S 

-s  -e  -J 


c 


h 


-r 


*ec 

d 

*i 

d 

CO  co  co  CO  co  co  CO  co  co  CO 

^ 

i. 

OOOOOOOOOO 

S^ 

S 

d 

5SS22S"*--:; 

Si 

d 

SgS?£8S38££S 

oooooooooo 

If 

^cO^^C^O^^i 

. 

41  L 

a 



(M 

1-2 

O2 

« 

d 

CO^^LOCOCO^GOOSOS 

ooaoooooGOoooooooOGO 

oooooooooo 

" 

A 

SSSSSS22D* 

H 

d 
t-t 

5  9  8  !;  SI  1?  §  S  S  S 

H 

^     1 

d 



1 

"** 

2 

X 

^ 

c 

a 

^    <N    <N    <N    <N    <N    <N    <N    <N    <N 

- 

d 

™555S5^5«^ 

d 

03 

o 

£' 

B'| 
£ 

G 

5*2SSSS««2 

,£2 

M 

^ 

05 

3 

-ooo»^-oo»e,oo 

^ 

P<rv 

0 

1--    CO    ^r    CO    CM    ^H    O^    CO   t^   *O 

^ 

OH 

><x><xxxxxxx 

i 

cocococococococococo 

C 

a 

a 

XXXXXXXXXX 

TABLES 


41 


<N<MC<l<NCq<N<N(MCOOO  COeOCOeO^^ThT*  N   O3   <N   C<1   CO   CO 

COtOCOCOOCOOCOCOCO  iOiOiOiQiO*O>OiO  1C   lO   »O   1C   »O   »O 

oddddddddo  OOoddddd  OOOOOO 

<NTHOCit^COlOCOi-<Oi  Ti<    Tj<    CO    C^J    i-H    O5    00    ?^  O    O>    00    l2    CO    »O 

<N     <N     <N     »-H     i-l     i-H     T-H     ,-1     l-<     O  l-ll-Hi-Hl-Hi-lOOO  i-HOOOOO 

OOCOTt<C^OOC»OCO'-iC5  l>iOCOOOOCOTtHi— I  b*.   *O   CO   ^H    00   CO 

pppppoooGooocb-  i>i^i>t^pcococo  i>  t^.  i>  i>  co  co 

dodddddoOO  C5OOOOOOO  OOOOOO 

00*O<NOO  C5(NrJHOoOC5O'— i  C<lrJ<cOGOC5O 

l^iO^cO(M_i-iO500l>iO  OCSOOt^COtOLOr^  00   t^   «O    >O   ^   •* 

^H'^J^^H^^HOOOO  OOOOOO  o'o  OOOOOO 

iOiOOI>l>OOO5OO'-i  OOOOi-i<NCOTt<  <N<NCO^^»O 

OOOOGOOOOOOOOOO1O5O5  OCOr^t^i>l^t>-l>  t>t^|>t>.t>.t>. 

o'odo'ooo'ooo  dooddood  ddddoo 

^   00  O   ^ 

COi— lOOOiOCOi— tOOOCO  l>O*O-*C<|i—iC>t^  -^COC^OO5t>- 

cococo(N(NC<ic^i-ir-<^-i  ^H^H^H^'^H'^'dd  ^HrHi-Ii-idd 

coi-iC5i>iocooao<DTt<  t^iocooooo^^H  <NOOOCOCOI-H 

(NfNi-Hi-lrHi-li-lppp  O«C<IC^(Ni-lrHi-l^-l  OOOSOOCi 

,H'     rn'     ^'     r-!     ^     ^H     ^H     ^     ^H     ,-H  ^H     ^     r-i     ^     rH     ^     ,-J     ^  ^-!     ^'    O    O    O    O 

§00  CO    *O  CC    i-H    O5    CO 

I>-  O5N-CO1^cO'-HC5t>  C^OOiOOCOO 

OJ(NT-lT-l,-l,-l,-!i-lOO  ^Hi-ti-H^-ti-Hi-HOO  i-Hi-HOOOO 

-^Tf»OCDt^l>GOOiO^H  Ot^-OCC5O5Oi-((N  i-tr-ifNCO^iO 

Pppppppp^H^H  ppppp^Hi-lT-l  OO5O3O5OO5 

^H^H^H^H_^H^^H^H^H  ^H     ^     _i     ^H'     _'     _<'     _•     ,H"  000000 

Ot»^i— lOO^Oi— i|>COO5  »-iOOCOC<IO5CO(NOO  CO»-iO5t^Tj<C^ 

lO   Tt<   Tt<   •*'   CO  CO  CO  Cfl   O*   i-<  ^COCOCOcicicit-H  (N i   CSJ [t-(   I-H   i-i   i-i 

g^8^c^8Si§S  SS8S^q?2^  ?2S^§S^ 

rJH   Tj<   TjJ   CO   CO  CO   <N*   (M    1-1    T-I  CO    CO   CO   <N    (N    <N    i-I   I-H  (N    IM'   <N    i-I   i-i   i-i 

OOt^COiOrC^rHcO^t1  lOiC^TjHcOC^i— i    Oi  U5iOCDCOCOiO 


xxxxxxxxxx  xxxxxxxx  xxxxxx 

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COCOCOCOCOCOCOCOCOCO 

xxxxxxxxxx 

^N  ^!M  r*le*  HM  "!N  -!«  "IN  -•«  w.«  WN 
COCOCOCOCCCOCOCOCOCO 


xxxxxxxx 

rt^  Hn  ">«  "ic«  H«  ^i«  Mk«  H« 
COCOCOCOCOC'SCOCO 


xxxxxx 

COCOCCCOMCO 


42 


TABLES 


r 


• 


f 


s  I 

PQ  •< 

$  § 


J 


is 


a    "    S 

CO       3       ^ 

111 


2w 

^^ 

d 
'1 

d 

£2  £2  £2  £2  £2  SJ  SJ  S  S  S5  £2  £2 

TT   "^   ^r  '^   "^    "^   ^r   ^r   ^r  ^r   ^r   ^r 

d  d  d  o  d   doddddd 

21 

*^ 

S 

d 

Tt<    l>«    i—  i    »O    OS     CO   CO   O    -"f    00    C^l    CO 
CO    *O    »O    TfH    CO      -*    TjH    rtl    CO    <N    <M    r-H 

^§ 

00000     0000000 

A 

d 

00   CO   Tt<    <M    C5     CO    O   00   CO   "*    rH    OS 
IO   iO   »O   »O   "f     CO   CO    lO   iO   »O   *O   rfl 

oddoo   d  d  d  d  o  d  d 

<ag 

H     1 

d 

l>   (N    t^   <N    lO     CO    I-H    CO    1-1    IO    O   CO 

•<*    TjH    CO    CO    <N     •^tl^COCO(N(NT-l 

<N 

PNd 

ooooo   ooooooo 

05 
>< 

•ij 

e 

d 

voiocot^t^     <£>Ir^OOOOO5Oi-i 

io>o>oioio    io^oio^o>ococo 
ooodd   ooddddd 

^ 

d 

t>.   i—  i    Tt<   !>•   O5     rfHGOi-H»OI^O5O 
OCOtOTfriCO     OiOiO-^CO(N<M 

oddoo   dodddoo 

H     - 

d 

ooo^cqoi    ooiocoT-HOscO'* 

OOOO05     00000000l>t>i> 
I-H   T-H    T-H   i—  i    O     OOOOOOO 

v 

H 

^  L 

d 

O500CDTF     O(MtO1^.00O5O 
OOOl>CO»O     1>CO»OTJHCOC<I(M 

T-n'oodd   odoodod 

03 

M 
<| 

d 

C^CO-^»OiO     iOCOt>-OOOOO5O 

ooooo   ooooooo 

d 

i—  I    O5    1C    i—  1    IO 

os  i>  >o  co  T—  i    T-nooii>coioco 

S 

<: 

d 

o 

si 

d 

lOOCOt^Oi     O   00    IO   T—  i    CO   i—  i    "3 
(MOt^^r-H     Ot^»OCOOCO»O 

OQ 

1 

fe"8 

-3 

a 

(N    »0   CO 
1>OOO5O'-H     CO   i—  I    CO   "3   CO   1>   00 

1 

ftf^ 

1 

t^.CO»OiOT^     COcOiO^COC^i-* 

g 
fi 

Inches. 

H|«  HS  «l=o  HS  HM    H|N  HS  «|«  HS  HH-  HS  H|« 

xxxxx  xxxxxxx 

(N(N(N(N(N     <N<N<N<N(N<N<N 

xxxxx  xxxxxxx 

H|M  ^!M  He*  HN  HN  ^l"  He« 
COCOCOCOCO     (NCN(N(N(N(N(N 

» 

TABLES 


43 


8M 


CO    CO    CO 

o'  d  d 


CO 


000000 


Ot^rtH 
<N    I-H    1-1 

o'oo 


<N    1-1    O   t^   »O   <N 
<N    O*    <N    .-i    i-H    T-H 

o  o  d  d  d  d 


O   00    1C 

•^    CO    CO 


$$: 

o  d  o  o  o  o 


OS   1> 

CO    CO 


ooo 


co  co  o  i>  ^  I-H 

(^     <^j     CNJ     ,_|     ,_,,_, 

ooo'ooo 


o  o  o 


O   I-H    ^ 

T^     TF     Tt^ 

o  o  o 


<M    <N    CO 

T^     T^     T^H 

o  o  o 


o>  o  co 

i— I    i— I     rH 

o  o  o 


000 


OS   CO   «M 

odd 


os  oo  oo 
odd 


O    CO    i—  I 
OO    GO    00 

o  d  d 


C5   t>-    lO 

l>    l>    l> 

o  d  d 


TjH    O    00 
•^    CO    <N 

odd 


»O    "*i    Tt< 

o  d  d 


CO   O   CO 
CO    CO    (N 

o  d  d 


o  o  o       o  o  o 


o  i— i  cq 
t>.  t-  l> 

d  o  d 


i-H    OS    CO 

l>   1C   T^ 

odd 


10     00     i-H 
t>-     CO     O 

O   O   O 


CO     Tt<     Tt< 
1C     TjH     CO 

O   O   O 


i-H     O     O  i-lrH^H^H 


<N    C5 

OS    rH 


COCOON 


XX  X       X XX  XX  X 

HC«    HC*    ^'^  HN    HN    HN     HW    -*i«     -HJM 

XXX       XXXXXX 


44 


TABLES 


TABLE  III 
COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOR  ANGLES 


COORDINATES. 


Size  of 
Angle. 

Xab 

•? 

Xbc 

+ 

XCd 

V 

Xde 

yae 

Xea 

yea 

8X8X11 

9.45 

9.45 

10.25 

17.53 

40.66 

23.78 

23.78 

40.66 

17.53 

10.25 

8X8X1 

8.66 

8.66 

9.25 

15.81 

37.55 

21.98 

21.98 

37.55 

15.81 

9.25 

8X8X1 

7.67 

7.67 

8.29 

14.01 

34.31 

20.30 

20.3034.31 

14.01 

8.29 

8X8X1 

6.80 

6.80 

7.20 

12.18 

30.57 

18.07 

18.07 

30.57 

12.18 

7.20 

8X8X1 

5.76 

5.76 

6.13 

10.29 

26.64 

15.87 

15.87 

26.64 

10.29 

6.13 

8X8X£ 

4.78 

4.78 

4.97 

8.36 

22.19 

13.20 

13.20 

22.19 

8.36 

4.97 

8X6X1 

7.95 

4.91 

7.47 

8.92 

30.49 

12.26 

19.70 

23.51 

15.39 

6.07 

8X6X1 

7.33 

4.47 

6.65 

7.95 

27.70 

11.19 

18.14 

21.68 

13.41 

5.42 

8X6X1 

6.36 

3.79 

5.86 

6.91 

24.77 

10.16 

16.67 

19.68 

11.65 

4.78 

8X6X1 

5.56 

3.27 

4.96 

5.87 

21.47 

8.81 

14.60 

17.30 

9.87 

4.05 

8X6X£ 

4.57 

2.66 

4.06 

4.79 

17.39 

7.45 

12.52 

14.76 

8.01 

3.33 

8X6X3^ 

4.08 

2.38 

3.58 

4.24 

16.00 

6.65 

11.24 

13.31 

7.06 

2.94 

8X3£X1 

6.76 

2.15 

4.50 

3.02 

20.88 

3.66 

12.61 

8.48 

13.71 

2.40 

8X3|Xl 

6.15 

1.73 

4.11 

2.70 

19.04 

3.46 

12  .41 

8.16 

12.17 

2.21 

8X3|Xf 

5.58 

1.43 

3.66 

2.35 

17.04 

3.19 

11.95 

7.68 

10.61 

1.99 

8X3^X| 

5.06 

1.19 

3.09 

1.98 

14.75 

2.77 

10.77 

6.92 

8.99 

1.69 

8X3^X1 

4.26 

0.93 

2.56 

1.62 

12.31 

2.38 

9.73 

6.16 

7.31 

1.41 

8X3£Xfk 

3.79 

0.85 

2.29 

1.46 

11.02 

2.17 

9.14 

5.86 

6.44 

1.27 

7X3|X1 

5.07 

1.88 

3.82 

2.95 

16.75 

3.58 

10.10 

7.81 

10.60 

2.26 

7X3|Xl 

4.65 

1.61 

3.61 

2.63 

15.35 

3.38 

9.89 

7.47 

9.41 

2.07 

7X3^Xf 

4.28 

1.40 

3.08 

2.32 

13.75 

3.09 

9.32 

7.01 

8.22 

1.87 

7X3^X1 

3.79 

1.16 

2.65 

1.98 

12.02 

2.66 

8.66 

6.46 

7.13 

1.60 

7X3£X£ 

3.27 

0.94 

2.17 

1.62 

10.05 

2.33 

7.06 

5.64 

5.68 

1.32 

7X3JXI 

2.56 

0.72 

1.70 

1.26 

7.90 

1.90 

6.44 

4.80 

4.34 

1.04 

TABLES 


45 


TABLE  III— Continued 
COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOB  ANGLES 


COORDINATES. 


Size  of 
Angle. 

Xab        yab 

Xbc 

¥ 

Xcd 

¥ 

Xde 

•? 

Xea 

yea 

6X6X1 

4.54J  4.54 

4.98   8.57 

19.10 

11.08 

11.08 

19.10 

8.57 

4.98 

6X6X1 

4.1l!  4.11 

4.45   7.64 

17.54 

10.21 

10.21 

17.54   7.64 

4.45 

6X6X1 

3.70  -3.70 

3.89   6.59 

15.86 

9.23 

9.23il5.86   6.59 

3.89 

6X6X| 

3.13    3.13 

3.35;  5.67 

14.00 

8.26 

8.2614.00 

5.67 

3.35 

6X6X£ 

2.55 

2.55 

2.75    4.61 

11.84 

7.08 

7.0811.84 

4.61 

2.75 

6X6X1 

2.00 

2.00 

2.11 

3.53 

9.40 

5.61 

5.61 

9.40 

3.53 

2.11 

6X4X1 

4.03 

2.28 

3.50 

3.82 

14.20 

4.56 

8.46 

9.23 

8.05 

2.58 

6X4X1 

3.58 

1.95 

3.23 

3.40 

13.06 

4.38 

8.30 

8.75 

7.23 

2.40 

6X4X1 

3.27 

1.70 

2.84 

2.9811.78 

3.99 

7.68 

8.05 

6.25 

2.12 

6X4Xf 

2.85 

1.40 

2.46 

2.53J10.40I  3.60 

7.09 

7.281  5.32 

1.84 

6X4X£ 

2.46 

1'.20 

1.99 

2.09 

8.74 

3.02 

6.06 

6.36 

4.33 

1.50 

6X4X1 

1.90 

0.88 

1.57 

1.60 

6.96 

2.47 

5.11 

5.21 

3.32 

1.18 

6X3^X1 

3.66 

1.77 

3.13 

2.89 

12.92 

3.45 

7.72 

7.12 

7.81 

2.08 

6X3|X| 

3.43 

1.58 

2.84 

2.61 

11.89 

3.24 

7.42 

6.80 

6.99 

1.90 

6X3^X1 

3.19 

1.34 

2.49 

2.26 

10.69 

2.94 

6.88 

6.24 

6.10 

1.68 

6X3^X| 

2.78 

1.13 

2.18 

1.95 

9.44 

2.68 

6.48 

5.80 

5.20 

1.47 

6X3^Xi 

2.35 

0.93 

1.80 

1.61 

7.98 

2.31 

5.78 

5.18 

4.24 

1.22 

6X3^X| 

1.89 

0.66 

1.40 

1.22 

6.32 

1.86 

4.81 

4.18 

3.26 

0.96 

6X3£XA 

1.571  0.59 

1.20 

1.06 

5.42 

1.64 

4.34 

3.82 

2.73 

0.83 

5X5X1 

3.06    3.06 

3.27 

5.78 

12.18 

6.90 

6.90 

12.18 

5.78 

3.27 

5X5X| 

2.81    2.81 

2.94 

5.2011.33 

6.44 

6.44 

11.33 

5.20 

2.94 

5X5X1 

2.40:  2.40 

2.64 

4.51110.32 

6.05 

6.05 

10.32 

4.51 

2.64 

5X5X1 

2.10;  2.10 

2.27 

3.84 

9.20 

5.40 

5.40 

9.20 

3.84 

2.27 

5X5Xi 

1.74:  1.74 

1.88 

3.16 

7.90 

4.69 

4.69 

7.90 

3.16 

1.88 

5X5X| 

1.27    1.27 

1.50 

2.41 

6.25 

3.88 

3.88 

6.25 

2.41 

1.50 

46 


TABLES 


TABLE  III— Continued 
COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOR  ANGLES 


COOKDINATES. 


Size  of 
Angle. 

- 

7 

Xbc 

yi>c 

Xcd 

- 

Xde 

yde 

Xea 

yea 

5X4X1 

2.56 

1.84 

2.47 

3.30 

9.59 

4.04 

5.70 

7.60 

4.99 

2.10 

5X4X1 

2.27 

1.58 

2.22 

2.88 

8.80 

3.80 

5.43 

7.07 

4.37 

1.89 

5X4Xf 

2.00 

1.35 

1.91 

2.46 

7.78 

3.40 

4.91 

6.34 

3.72 

1.63 

5X4X£ 

1.65 

1.12 

1.60 

2.05 

6.68 

2.99 

4.39 

5.60 

3.06 

1.37 

5X4X1 

1.33 

0.91 

1.21 

1.58 

5.30 

2.35 

3.50 

4.56 

2.33 

1.04 

5X31X1 

2.44 

1.48 

2.19 

2.52 

8.77 

3.02 

5.19 

5.96 

4.89 

1.68 

5X3|X| 

2.21 

1.31 

1.96 

2.24 

7.94 

2.80 

4.90 

5.60 

4.27 

1.51 

5X3^X1 

1.89 

1.03 

1.72 

1.88 

7.06 

2.59 

4.63 

5.06 

3.63 

1.33 

5X3^X1 

1.63 

0.84 

1.43 

1.54 

6.02 

2.23 

4.06 

4.40 

3.00 

1.11 

5X3^X1 

1.31 

0.69 

1.10 

1.21 

4.84 

1.80 

3.37 

3.72 

2.30 

0.86 

5X3iX^ 

1.11 

0.56 

0.94 

1.02 

4.14 

1.57 

2.98 

3.22 

1.93 

0.73 

5X3Xif 

2.14 

1.04 

1.82 

1.73 

7.52 

2.10 

4.53 

4.30 

4.46 

1.24 

5X3X1 

2.08 

0.97 

1.71 

1.62 

7.18 

2.01 

4.40 

4.16 

4.17 

1.17 

5X3X1 

1.85 

0.84 

1.50 

1.41 

6.34 

1.83 

4.12 

3.88 

3.56 

1.03 

5X3X1 

1.59 

0.67 

1.24 

1.16 

5.42 

1.60 

3.73 

3.47 

2.92 

0.86 

5X3X1 

1.23 

0.45 

1.00 

0.87 

4.35 

1.36 

3.29 

2.86 

2.24 

0.70 

5X3X& 

1.11 

0.40 

0.82 

0.78 

3.75 

1.13 

2.80 

2.64 

1.90 

0.57 

4iX3Xii 

1.75 

1.02 

1.57 

1.72 

6.24 

2.00 

3.67 

4.00 

3.62 

1.16 

4|X3Xf 

1.71 

0.97 

1.46 

1.60 

5.95 

1.90 

3.53 

3.87 

3.38 

1.08 

4|X3Xf 

1.42 

0.81 

1.29 

1.38 

5.31 

1.77 

3.38 

3.62 

2.88 

0.96 

4^X3X1 

1.27 

0.64 

1.09 

1.13 

4.54 

1.56 

3.04 

3.17 

2.37 

0.81 

4|X3Xf 

1.03 

0.51 

0.84 

0.88 

3.69 

1.27 

2.57 

2.70 

1.83 

0.63 

4JX3XA 

0.92 

0.44 

0.70 

0.75 

3.20 

1.09 

2.22 

2.36 

1.54 

0.53 

TABLES 


47 


TABLE  III — Continued 
COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOR  ANGLES 


COORDINATES. 


Size  of 
Angle. 

Xab 

+ 

yab 

Xbc 

ybc 

Xcd 

ycd 

+ 

Xde 

+ 

Vde 

Xea 

+ 

yea 

4X4XM 

1.57 

1.57 

1.70 

2.99 

6.28 

3.57 

3.57 

6.28 

2.99 

1.70 

4X4X1 

1.49 

1.49 

1.61 

2.82 

6.06 

3.47 

3.47 

6.06 

2.82 

1.61 

4X4X1 

1.34 

1.34 

1.37 

2.42 

5.44 

3.09 

3.09 

5.44 

2.42 

1.37 

4X4Xi 

1.12 

1.12 

1.13 

1.99 

4.74 

2.71 

2.71 

4.74 

1.99 

1.13 

4X4X1 

0.86 

0.86 

0.91 

1.54 

3.86 

2.28 

2.28 

3.86 

1.54 

0.91 

4X4Xi 

0.58 

0.58 

0.62 

1.03 

2.76 

1.66 

1.66 

2.76 

1.03 

0.62 

4X3IXM 

1.54 

1.28 

1.51 

2.30 

5.74 

2.65 

3.24 

4.96 

2.96 

1.36 

4X3^Xf 

1.50 

1.22 

1.41 

2.16 

5.45 

2.54 

3.12 

4.77 

2.74 

1.28 

4X3|Xf 

1.24 

0.98 

1.26 

1.83 

4.96 

2.40 

2.98 

4.33 

2.36 

1.14 

4X3^X£ 

1.05 

0.85 

1.04 

.52 

4.24 

2.08 

2.60 

3.80 

1.93 

0.95 

4X3^X| 

0.83 

0.65 

0.83 

.18 

3.47 

.73 

2.19 

3.12 

1.50 

0.75 

4X3IX& 

0.72 

0.57 

0.70 

.01 

3.05 

.53 

1.93 

2.80 

'1.28 

0.64 

4X3XH 

1.42 

1.01 

1.31 

.70 

5.07 

.87 

2.87 

3.72 

2.85 

1.06 

4X3X1 

1.35 

0.93 

1.25 

.59 

4.86 

.83 

2.83 

3.59 

2.57 

1.01 

4X3X1 

1.13 

0.75 

1.13 

.36 

4.38 

.75 

2.76 

3.33 

3.28 

0.91 

4X3X1 

1.00 

0.62 

0.92 

.10 

3.76 

1.50 

2.41 

2.89 

1.87 

0.75 

4X3Xf 

0.83 

0.47 

0.72 

0.86 

3.12 

1.25 

2.05 

2.43 

1.47 

0.59 

4X3Xi 

0.57 

0.37 

0.49 

0.62 

2.26 

0.89 

1.49 

1.89 

1.01 

0.40 

3£X3£XH 

1.22 

1.22 

1.24 

2.27 

4.53 

2.48 

2.48 

4.53 

2.27 

1.24 

3|X3|Xf 

1.13 

1.13 

1.19 

2.13 

4.35 

2.43 

2.43 

4.35 

2.13 

1.19 

3^X3|Xf 

0.93 

0.93 

1.04 

1.79 

3.91 

2.27 

2.27 

3.91 

1.79 

1.04 

3^X3|X£ 

0.80 

0.80 

0.86 

1.48 

3.40 

1.98 

1.98 

3.40 

1.48 

0.86 

3£X3|Xf 

0.65 

0.65 

0.68 

1.16 

2.87 

1.68 

1.68 

2.87 

1.16 

0.68 

3£X3^X* 

0.44 

0.44 

0.47 

0.79 

2.06 

1.24 

1.24 

2.06 

0.79 

0.47 

48 


TABLES 


TABLE  III— Continued 

COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOB  ANGLES 

Y 


COORDINATES. 


Size  of 
Angle. 

Xab 

+ 

Vab 

+ 

Xbc 

Vbc 

Xcd 

ycd 

Xde 
+ 

yde 

Xea 

+ 

yea 

3£X3XH 

1.11 

0.90 

1.09 

1.63 

4.06 

1.79 

2.24 

3.37 

2.20 

0.97 

3^X3Xf 

1.05 

0.83 

1.03 

1.52 

3.89 

1.74 

2.19 

3.23 

2.06 

0.92 

3^X3Xf 

0.91 

0.75 

0.91 

1.34 

3.51 

1.62 

2.07 

3.04 

1.76 

0.82 

3£X3X£ 

0.82 

0.61 

0.75 

1.08 

3.10 

1.41 

.82 

2.61 

1.48 

0.68 

3^X3X1 

0.61 

0.45 

0.60 

0.83 

2.50 

1.20 

.57 

2.17 

1.11 

0.54 

3|X3X| 

0.45 

0.34 

0.42 

0.59 

1.83 

0.88 

.16 

1.64 

0.77 

0.37 

3|X2|XH 

0.85 

0.59 

0.85 

0.98 

3.23 

1.18 

.90 

2.21 

1.84 

0.65 

3£X2^Xf 

0.80 

0.51 

0.80 

0.91 

3.04 

1.12 

.87 

2.13 

1.69 

0.62 

3£X2|X£ 

0.70 

0.45 

0.67 

0.78 

2.67 

1.00 

.72 

2.00 

1.39 

0.52 

3£X2£X! 

0.61 

0.34 

0.53 

0.60 

2.24 

0.84 

1.47 

1.67 

1.11 

0.41 

3|X2£Xi 

0.41 

0.22 

0.38 

0.41 

1.62 

0.64 

1.16 

1.28 

0.75 

0.29 

3X3X1 

0.66 

0.66 

0.74 

1.29 

2.65 

1.53 

1.53 

2.65 

1.29 

0.74 

3X3X£ 

0.55 

0.55 

0.63 

1.06 

2.36 

1.40 

1.40 

2.36 

1.06 

0.63 

3X3X1 

0.50 

0.50 

0.47 

0.85 

2.02 

1.12 

1.12 

2.02 

0.85 

0.47 

3X3X1 

0.29 

0.29 

0.35 

0.55 

1.43 

0.89 

0.89 

1.43 

0.55 

0.35 

3X2iX& 

0.59 

0.43 

0.59 

0.81 

2.26 

1.00 

1.32 

1.82 

1.16 

0.51 

3X2|X£ 

0.55 

0.41 

0.54 

0.74 

2.10 

0.94 

1.25 

1.73 

1.05 

0.47 

3X2^Xf 

0.48 

0.30 

0.42 

0.56 

1.77 

0.78 

1.06 

1.41 

0.83 

0.37 

3X2£Xi 

0.33 

0.21 

0.30 

0.40 

1.32 

0.60 

0.83 

1.-12 

0.57 

0.26 

3X2Xi 

0.47 

0.27 

0.45 

0.47 

1.76 

0.59 

1.10 

1.16 

0.99 

0.33 

3X2Xf 

0.40 

0.21 

0.35 

0.37 

1.44 

0.49 

0.94 

1.00 

0.77 

0.26 

3X2Xi 

0.30 

0.14 

0.26 

0.26 

1.11 

0.39 

0.80 

0.80 

0.55 

0.19 

TABLES 


49 


TABLE  III— Conduded 

COORDINATES  OF  SECTION  MODULUS  POLYGONS  FOB  ANGLES 

Y 


COORDINATES. 


Size  of 
Angle. 

Xab 

+ 

yab 

+ 

Xbc 

ybc 

+ 

Xcd 

yea 

+ 

Xde 

+ 

yae 

Xea 

+ 

yea 

2|X2iX£ 

0.38 

0.38 

0.40 

0.71 

1.48 

0.84 

0.84 

1.48 

0.71 

0.40 

2iX2|Xf 

0.30 

0.30 

0.33 

0.56 

1.29 

0.75 

0.75 

1.29 

0.56 

0.33 

2£X2£Xi 

0.22 

0.22 

0.23 

0.39 

0.97 

0.57 

0.57 

0.97 

0.39 

0.23 

2|X2£Xi 

0.11 

0.11 

0.13 

0.21 

0.57 

0.34 

0.34 

0.57 

0.21 

0.13 

2|X2X£ 

0.34 

0.27 

0.34 

0.47 

1.25 

0.52 

0.73 

1.01 

0.68 

0.28 

2£X2Xf 

0.28 

0.19 

0.28 

0.36 

1.10 

0.48 

0.69 

0.88 

0.54 

0.24 

2|X2Xl 

0.22 

0.15 

0.19 

0.25 

0.82 

0.35 

0.52 

0.68 

0.38 

0.16 

2±X2Xi 

0.11 

0.07 

0.11 

0.13 

0.47 

0.22 

0.33 

0.41 

0.20 

0.09 

2£XHXA 

0.24 

0.10 

0.18 

0.17 

0.79 

0.22 

0.50 

0.47 

0.44 

0.12 

2^XUXi 

0.20 

0.08 

0.15 

0.14 

0.67 

0.19 

0.45 

0.42 

0.36 

0.10 

2ixi*x& 

0.16 

0.06 

0.12 

0.11 

0.54 

0.16 

0.40 

0.37 

0.28 

0.08 

2iXl£Xf 

0.26 

0.16 

0.22 

0.25 

0.87 

0.26 

0.48 

0.54 

0.54 

0.16 

2JXUXI 

0.21 

0.12 

0.19 

'0.20 

0.75 

0.25 

0.45 

0.48 

0.42 

0.14 

2JXUXJ 

0.16 

0.09 

0.14 

0.14 

0.57 

0.19 

0.38 

0.41 

0.30 

0.10 

2|XHX& 

0.12 

0.06 

0.11 

0.11 

0.45 

0.16 

0.32 

0.32 

0.23 

0.08 

2X2X^ 

0.21 

0.21 

0.22 

0.40 

0.82 

0.45 

0.45 

0.82 

0.40 

0.22 

2X2X1 

0.19 

0.19 

0.20 

0.35 

0.75 

0.42 

0.42 

0.75 

0.35 

0.20 

2X2Xi 

0.13 

0.13 

0.15 

0.25 

0.59 

0.36 

0.36 

0.59 

0.25 

0.15 

2X2X1 

0.08 

0.08 

0.08 

0.13 

0.35 

0.20 

0.20 

0.35 

0.13 

0.08 

50 


TABLES 


e 


- 


a    -* 

•2      £3 

Q    <J 


o 
o 

0 

k  o 

o 

0 

o 
o  ) 

(•& 
o 
o 
o 

ty-\ 
o 
o 

0 

g  § 

H    O 

$& 


» 


•*f  00 
1C  CO 
(N  (N 


ooosoooscocooocooo 


cS 


t^  10  GO  co  oo  o 


COrHlOrHlOOOOrH 


^^  CO  CO 
CO  (N  rH 


1C  t^  CO  "* 


CO   1C 

Oi   00 


1C  TH  1C 

Oi  O  O 

rH  TH  O 


Oi   CO 
OS    00 


Oi  rH  CO  CO  "*  rH  O 


10  CO 
CO  1C 


OO  Oi  O 


O  CO  CO  00  rH 


Oi  00  00 


CO  <N  TH  TH  O  Oi 


OS  O3  1C 

CO  00  l> 


CO  CO 


(M  00  CNI  t^ 

rH  CO  Oi  Oi 


t>*  CO 


CO   i 


lO  Oi 

rH  O 


S!  ^  fS 

t"*  i-H  tO 

oi  OS  00 


o  co 
CO   »C 


CO  Oi  CO 

t^  rH  CO 


OS 


00 


CO  CO  1C  1C 


(N I  1>  TH 

Oi  GO  OO 


00  C^  CD 
O  iO  Oi 

t>^  CO  id 


Oi  Oi  00  OO 


OO  OO  00  00  00 


00  00  00  00 


TJH  CO  CO 
00  00  00 


lOiOiOiOiOiOiOiOiC 


O  O  O  O  TH  rH 


Oi  Oi 
CO  CO 


CO  CO 


O5 
CO 


Oi  Oi 
CO  CO 


o    o 
CO  CO 


1C  1C  1C 

CO  co  co 


00  00  Oi  Oi  O  O 


co  co  CD  CD  CO  CO 


CD  CO  CD 


H5   SfcH-SS-^^HS, 


oo 


TABLES 


51 


CO  00  00  Oi  OO  Is*  CO  ^4  ^O  ^H 


00  T*<  CO  OS 

rH  rH  rH  O 
CO  tO  Tt*  CO 


rH  ^  00  O> 


OSOGOCSb-COOT^GOC^ 
CO   to   to   CO   b*   GO   OS    OS   OS   ^^ 

tOTfCOC<|rHOO3GOb»b- 


OS  p  OS  GO  GO 
CO  M*  CO  CO  CO 

rH  O  OS  GO  b- 


<NrHOOT^rHtOb-p<NrH 


tO     CO    "*     CO    00     tO    rH 


OCOOCOTt<Tt<TfCOrHCO 
COC^rHOOOSOOb-COCO 


O   CO   O 


CO   00   00   CO 


Sgol 


O     tO     rH     IO     GO 


rH     CO     CO     GO     CO     tO 


b-   OS    O   <N    CO 

OS   00   00   b»   CO 


O   CO   OS   (M   •*   <N 


CO     rH     OS     CO 

tO    GO    CO    CO 
b*   CO   CO   to 


tO    00    rH    CO 

3    r2    O     O 


CO     CO     rH     O     CO 


CO    *-O   4s*    O^ 

os  oo  b-  co 


rJH   GO   OQ   CO 

O  OS  OS  00 


O  O   OS  00   CO 


CO     GO     rH     IO    b- 

IO    GO    <N    to    GO 

b»  CO  CO   to  TJ< 


OS  OS  00  b- 


Os   OS   OS   OS 

rH     IO    OS    CO 

CO    to   rti   rfi 


CO    CO    Tt< 

to  os  co 

OS   00   00 


•^ 


(N   CO 
tO   rj< 


OS    GO    CO    "^    C^    GO 
CO   CO   GO   CO   GO   C^ 

CO   00   b»   b-   CO   CO 


Os  OS  CO 


CO  CO  to  ^ 


CO 


00 


b-  CO  CO 


O    C^    O   CD 
tO   O   to   OS 

06  oo  i>  cd 


O   Tt<   CO 

CO    b-     rH 


W    00    ^ 
b^  cd  cd 


CO 


(M  t>. 

tO  OS 


00   00  b- 


1C 
<N 


CO  CO  to 


<N    GO 

CO    CO 


(M  CO 

GO  CO 


CO     (N     rH     CS 

CO    GO    CO    b* 


00 


co  co  to  to 


00    00    GO    GO    GO    00 


o  o 

X     X 


0000 


00000 


OOOOOOOOOOOOOOb-b- 


T*Tj<Tt(Tt<Tj<'HH'<tlTi< 

b-b-b-b-b-b-b-b- 


b-  b-  b-  b- 
b-  b-  b-  b- 


b-  b-  b- 
b-  b-  b- 


^^   ^^    lO   to   CO   CO    CO   b* 

OOOOOOGOGOGOGOGO 
COCOCOCOCOCOCOCO 


b-  b-  b-  b-  b- 


b-  b-  b-  b-  b- 

CO    CO    CO    CO    CO 


CO   GO    CS    OS 


OSOSOSOOOrHrH(N(M 


CO  CO 
CO  CO 


00000 
b-  b-  b-  b-  b- 


tOtocOb-OCOOOOs 

OS   OS   OS   OS   OS   OS   OS   OS 
tOtOtOtOiOiOtOtO 

OOCSOrH<M<MCOT*l 

to  to  CO  CO  CO  cO  CO  CO 
IQ  iQ  1C  t.t  1C  tC  to  to 


to   CO 

CO   CO 
tQ   tQ 


cOCOb-b-GCGOOSO 


§cOcOcOcOcocOcOcO 
cO   CO   CO   CO   CO   CO   CO   CO 


b-   GO   CO    OS   O 


sBn.se 


CO    CO 
CO   CO 

HSne. 


52 


TABLES 


O 

o 

0 

1  o 

0 
0 

o 
o  ) 

(-&• 

0 

o 

0 

o--\ 

0 

0 

o 

•8  a 

8  > 

§  > 

$  8 

•~i 

o    5 


3    ° 

«3     t 


1 1 


4-i 

02     S 

a  -H 

i* 
<  I 

"S 

i 
"rt 

'3 

o" 

w 


~         rS 


^  CC5  rH 

t>.    Ci    rH 

CO    <N    <M 


00    »C    O 

l>    O    CO 

(N     CM     rH 


00   (N    CO   Ci 


i 


1Q    OO    O^ 

Ci   <M   IO 

rH    rH    O 


OpppfNCiCOOO 
Ci<NiOI>O<NidNi 


rng 


CirHOOCirHCNrHi-l 

O^COCiOQcOCiOQiO 


CO   CO   Ci   rH   O  CO 


«.-  rH  10  00  <M  CO  Ci 
00  00  1>  CO  CO  IO  rH 


00  O  CO  CO  CO  CN  rH 


rH  GO 
Ci  00 


COI>.rHlOCiCOCOO 

OOI>l>COiOiOrHrH 


00    00 


CO    CO 


tO   O 

id  id 


CO    rH    CO 

CO   CO   1C 


b-   10   <M 

rH   O  CO 

l^-  t>-  CO 


Ci    IO    rH 
rH    t^    CO 


tO   Ci 
00    CO 

r}<   rH 


rH     rH     rH    O     Ci 

CO   <N   00   rH   Ci 


O   CO   (N 
!>•   CO   CO 


CO   CO 

IO    rH 

rH   rH 


£8°* 


00 


CO   <N_   00 

o  cd  id 


^^    rH    |>»    CO    O^ 

id   id  rH   r)H   CO 


§00   CO 
O  CO 

CO   CO   (N 


i_Q    ^^    iQ 
C3   Ci   IO 

CO   IO  IO 


2S3 


(N    <M    (M 

CO   Ci   IO 


rH   rH   CO   CO   <N 


rH    rH    O 

CO   OO   00 


pop 

00    00    00 


pop 

00   GO   OO' 


p  CO  CO 
t^  i>  t^ 


00   00   Ci   Ci   O 


CiOOrHrHCQCNCOrHrH 


s 


CO 
CO 


CQ   CO   rH   >O  CO 


rH    rH    (N 

CO   CO   CO 


COrHtOcOl>OOCip 

gO  O  O  O  O  O  rH 
CO  CO  CO  CD  CO  CO  CO 


00  Ci  p 
l^ 

10 


g^ 


rH  OQ  CO  rH  tO 


1>  t^  ^ 

IO  IO  to 


rHH 


TABLES 

O    i-H    00    IO    <N    CO  CO    CO    I-H    CO 


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C0 


O5    ^H    CM    O5    O5 

CM   N-   I-H    rt<   CO 

CO    1C   »O   Tf    CO 


O5   "tf1    OO    O   CM    CM        iOrHrt<00 
•^    ^H^    "^    CO    CO    CM        CO    CO    CM    CM 


CM   00 

S   CO 


O5    I-H 
bi  c^ 


b=S£§3 


CO    b-    O5    CM    CO 

§Tt^    CO    CO    b* 
iO    Tt<    T^    CO 


00     10     0 


b-    i-H         O5    CO    CO    CO 


t^    C5    CM    Tf<    lO 

O5   CO   00   C5   CO 


b-   CO   iO   IO        iO   iO 


CO    00    -^    O5 
rt<'   CO   CO   CM 


COCOCOCMCMCM        CMCMCMi-n 


co  i—  i 

00   CM 


10 

iO 


00 


lO   O   lO    O5   Tf 
U9  »O  "4<  CO  CO 


S  S 


CO    O 
CO   CO 


CO   CO   CM        CO   CO 


IO    I-H    tO    O5        *O   CO   CO    CO 
O5COCMOO        b-Tfii-Hb- 

CM    (N    CM    I-H        CM   CM   CM    T-H 


CM    CM    O    00    CO 
CO    05    10   O    CO 

r^    CO   CO    CO    CM 


3J  co  o  oo 

•^    I-H    00    •* 
CO    CO   CM    CM 


S8 

CM    I-H 


CM   C<1   CM   I-H 


O     T-H     ^H     O 

CM    CO    O    "^ 

co  id  10  -^ 


co  oo 

CM 


O5 
CO   CM 


t^  co 

CO    CO 


O   b^ 
CM    CM    <N    T-H 


iO    CM   S    CO 
CM    CM    TH   T-I 


S55 1:3 


•^    O   CO    i—  i 
CO    (N    t^    CO 


r>-  o  1-1  co 

10    CM    00    r^ 


10    lO 
10   CM 


iO    T^ 
O5   CO 


COCOCM        COCOCOCMCM        COCMCM<Mi-Hi-H 


i— i    CO    -^    tO 

Tj^     T-H     CO     10 

CM    CM    T-H    T-H 


Tt«     i— I     CO     i-H 

CO    I-H    iO   O 

IO     IO     Tt<     ^f 


OS     00     CD     Tt<     i-H 
CO     O5     lO     r-H     1> 


O    CO 

t>  co 


05 
O5 


COCOCOCM        COCOCOCMCM        CMCM<MCMT-HT-H        CMCMT-Hi-H 


b-  CO  CO  CO 
CO  00  CO  CO 


O  O5  O5  CO  CO 
OO  00  GO  GO  00 


b*-  t^-  CO  cO  CO 

CO  CO  00  CO  CO 


p  p 

CO  00 


p  p 

co  oo 


GO  p 
CO  CO 


oo  co 
b-  b» 

oToT 


CO   CM   <N    T-H 


CM    <N    CM    CM 

CO    OO    CO    CO 


^HH  Tt^  CO  CO  CO 
CO  GO  CO  00  CO 


10  to  to  to  to  oo  oo 


CO    CO 

t^  t^ 


CO   CO  CO  CO  CO 


00   CO    CO    CO 

b-  b-  b-  b- 


CO    CO    CO    CO    CO 


00  00 
CO  CO 


00    00    00 

CO    CO    CO 
l>   t>-   t^- 


O   I-H 


00 
!>•    t^ 


^H    CM 
00 


CM    CO 
00 


O    T-I    CM    CO 

o  o  o  o 


CMCMCMCM        T-I    i— I    CM    CM    CO        CO'HHiOiOtO        COCOb-OOO5O 


O5    Oi    O5    O5    O5 

CO   CO   CO   CO   CO 


b-    b-    b-    b-        CM    CO 


»o  »o  »o  10  *o 

co   CO   CO   CO   CO 


i-H    CM 

CO   CO 


o  o  o 
b»  b-  b- 

CO    CO    ^JH 


CO  CO  CO  CO 
CO   CO   CO   CO 


p    T-H    CM    CO 
CO   CO  CO   CO 


CO   l>   CO   O5   O 


l>  b-  t> 

CO   CO    CO 


CO    CO 
CO   CO 


CO    '^ 

CO   CO 


CM    CO   -<tfi    tO 

Tji     Tt<      ?*'     Tji 

CO   CO   CO   CO 


tOcOOb-        OOO5Oi-HCM 


b,  b,  b-  b, 

CO  O  CO  CO 


CO   CO   CO   CO 


33 


iO  CO  b-   b-  00  O5  O 
CO  CO  CO   co  CO  cO  CO 


l-H  <N 

CO  CO 


b-  oo  05  o 
co  CO  CO  co 


60 


TABLES 


o 

u 

Q 

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3    I 


bfi 
<U 
HH 


31 


o    § 
w   o 

II 

w    ' 

H     73 

UJ 

Q 

K 


II 


^   tf 


5  p 


I  s 


•t 


11  § 

•2     p 


cs 


O 

o 

0 

/ 

o 

V 

0 

0 

LI 


o    > 


HN 


§ 


1-1    (M    <N    CO    CO    CO   Tji    CO*CO    CO 


OiOOOi—  iCOCOCOi—  i 


»OO»OOiOOOi—  iCOiO 
OiCOCOOCOCOOCOCO 


iOOOOC^COCOCO(M'-i 
(Nl>COOOCOGOCOOOCO 


000 

O^    Oi    QJ 


(Mi—  lrHOOO001>t^CO 
•^•^TF-^^COCOCOCOCO 

oooooooooooooooooooo 


OOOOOOGO0000001>1>1> 


(N 


~S      d 


TABLES 


61 


COi-HVOCMcOOiHCO 


cJS? 


CM  CM  CM 

00  l>  CO 


CO 


tO  O5  CO 

CO  00  rH 

os  oo  oo 


CO  CM  »O  1>  OS 

CO  CO  00  O  CM 


CO  CO  OS  CM  Tfl 

Tj<  CO  CO  CM  rH 


CO  CM  OS  l>  CM 
to  CO  CO  t>-  00 


O  0^28 


CM  rH  OS  CO 

cO  cO  to  **^ 


i>  o  co  to  oo 

O  O  OS  00  Is*- 


CM    "* 
(M    i-i 


CO  b-  rH 

os  oo  oo 


00  CM 

00  CM 


rH  t^»  00  l>»  OS 

O  OS  00  00  t> 


CMt>-CMl>OrHrHrHOOOCO  rHCOOSCOtO 

5  CM  ^^  CO  OO  OS  CO    CO  CO  '^  OS  CO 

:>  OS  00  b-  CO  to  to   00  00  l>  CO  CO 


tO    CO    CM        COOcOrHCO 


O  CM  to 

CO  CM  rH 


N.  rH  tO  OS 

to  to  ^  eo 


b-  t>.  IO  "*  CM  CS 

CM  tO  00  rH  TP  CO 
CM  r3  O  O  OS  00 


to  oo  co  co  oo   oo  cs 

OS  rH  ^^  CO  OO    OS  ^^ 


l>  CO  p  CO  p  CM  CO   OOCSCO 


2§g 


CO     rH     OS 

to  06  o 

b-  CO  CD 


rHCOCOCOCMCSl>CM        OOCMCOCMCO 


$  s  s 

^^    Cv    CO 


SBSo1^ 


"^  «  * 

00   00   l> 


gg^ts 

CO   to   to 


Tfi    CO    00 

CO    to   to 


_     rH     rH  O 


CO     rH 

co  co 


OS  OS 


o  co  to  to 


CM    t^ 

OS    00 


co  oo 


CM    CO 

•^    CO 


§Sc3 


co  oq 

CO    CM 


t^   CM 

co  os 


00   l> 


CO   to 


CO   CO 


CO     rH 

00   00 


CO    O 

O     rH 


Ttl     i-H 

CO   1-1 


OS     10     rH 

iO    O    to 


Oi   ^^ 

co  co 


t^    t^    00 

O    CO   CM 


tO      Tj< 

O  CO 
CO    CM 


t^  CO  C0 


O   to   to   iO 


rH     CO 

00    l> 


O   CO 
CM    b- 


iO   CO 
(N   1> 


CO   CO   to 
CM    l>    CM 


t^  CO   CO   to   iO 


OS    CM 
to    CM 


S§3 

CM    CM 


CO    to   to 


O    CO    CM 


£3§ 


to    O 

co  co 


^8S 


COCOCOCMCM        tototo 


S3 

to  to 


t^   CM    00 

TjJ     TjJ     CO 


co  oo 
co  CM' 


CO     "^     rH     OO 


rH    00 

CO   CM 


CM   CM        to   to 


COCMrHOOOt^cOtOtO 


88 


t>.  l>  N.  cO 

oo  oo  oo  oo 


O  OS 
30 


CO  CO  CO 

oo  oo  x 


CO  to 

8*8 


COCMrHrHOOOSOO 


00  00  00 


CO    CO 
OO    00 


"^  CO  CO 
OS  OS  OS 


CM  CM 

OS  OS 


rH  rH     CO  CM  rH  O  O 


OS  OS 


oo  oo  t^  i>  b*.  co 

oo  oc  oo  oo  oo  oo 


OSOOOOOrHrH  OOO 


o  o  o 

00    00    00 


CS   CS 

cs  oi 


p  o  p  p  o  o 

Tf     Tti     Tt^     ^'     TjJ     T*H 


c  o  o  o  o 


^t^N-t^r^t^i^t^t^ 

CO   CO    CO   CD   o   CO   CO   CO   CO 


CM    CM    CM 


I>OCOCOC'CSCSOSOO        COl>OCC5OrHCMCO 


05  05  C5  05  C 

co  co  co  co  c: 


05    OS 

co   co 


CM  CM  CM  CM 

CO  CO  CO  CO 


t^.  I> 

CM  CM 


OOrHrHi— (CMCMCMCOCO^ 

cO  cO  cO  cO  cO  cD  cO  cO  cO  CO  CO 

CD  CD  CD  CD  CD  CD  CD  CO  CD  CD  CD 


tOt>OOCSOrHCMCO 


cste          r*o          _to 

-  r.   — —  r;—  — —  u-a:   *  — •  —  D 


CO   CO   CO   CO 


88 


62 


TABLES 


fc 

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o 

g1 

S 

o       O< 

S 

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o 

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0 

®  i 

Q 

J    2 

1 

^3        II 

g 

I     °  ! 

§ 

TJ             53 

o^       c3 

«          i    1 

HH 

CD                     -2      ~ 

H 

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H 

q 

o 

o 
Q 

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p 

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*+-  1 

1 

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1 

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E-i 

42 

J 

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c? 

1 

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0 
O 
0 

H 
hn 

« 

rn 

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PQ 
«1 

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•g 

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tf 

t-'oo        - 

g 

1 

0 

CD 

So 

0 

£ 

W 

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H 

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H 

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0 

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3  c 


EJ 

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co  os 


O    CO 
iO     CO 


o  ^ 

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CO  CO  CM  CO  to  CO  »O 
00  O  to  OS  CO  l>  rH 


oo  co  co 

CO  CO  to 


O  OS  CO  rH 


CO  *O  CO  ^^  OS 
iO  r}H  rtt  CO  CM 


X  OS  OS  O  00 

•^  t^  O  rfi  i— I 

CO  ^O  ^O  ^t1  »O 


c^?3 


rH   CO  O  1C  OS  CM 


OS  rt<  O  »O 
CO  CO  CO  CM 


•CM  OS 

OS  CM 

10  10 


CO  CO  T-H  OS  00  CO  00 

CO  CM 

CO  CO 


CO  T— I  CO  T— I   rH 


CO  CO  CM  CM  CM 


'o 


COCOCO     COCOCOCMCM 


OSCOOO     OOOiOT-HOCO 
1>-COOO     iO   i— I    00    >O   r- 1   00 

COCOCM     COCOCMCMCMi-i 


CM  X 

CO  ^H 


CO  CM  CM  CM 


CO  CO  TJH  CO 

X  TfH  O  CO 


o  F:  ^  ^  oo  to 

CO  <N'  CM  CM  rH  rH 


rH  10 

co  o 


CM  CM  CM  CM  rH  rH 


O  OS  OO 

"H^  CO  CO 

OO  00  OO 


OS  OS  OO  I> 


t^  1>  1>  1>  i> 


OS     00     00     00       rH     rH     CM     CM     CM     CO 

t^  l>  I>  t^  1>  N- 


OS   Os   Os   OS 
b»  IN.  l»»  IS 


CMCMrHrH       OSOrHCMCOCO 

CO  CO  co  CO  to  CO  co  CO  CO  CO 


X  00  00  00 
CM  CM  CM  CM 


co  co  co  co  co  co 


t^l>0000     tocOOOOrHCM 


HH  H«  HS 


H|«  HH 


TABLES  63 

00  CO   O   to   C5   C^  Oi   CM   HI   b»   cO   tO  CM   tH   CM   TH   O   cO  TH   TH   rj<   IO   ^    TH   Oi 

b-  T-H  to  06  T-I  to  d  to  O3  co  N.'  IH  co  b^  TH  to  03  <N  CM  co  d  Tji  06  CM"  to 

CO    CO    tO    Tt<    rt<    CO  CO    to    «tf    TJH    CO    CO  CO    to    IO    T£    CO    CO  CO    to    >O    35    CO    CO    CM 

»H~~cN    O    O3   b-   CO  CO   i— i    O3   b-   CM    00  O   ••*    00    O    IH    (N  O3   to   O   •*    00    TH   CO 

s'gS388  SS3838  ss$3$5c3  £8S3£g3 

OO    CM    -^    b~    b-    b-  tOO3CO«OtOtO  CO    O    >O    i— i    IO   b*  iH    TH    O    O3    CO    <H    O3 

StO    O3    CO    b»    iH  O    tO    iH    CO    iH    CO  b-    CM    CO    TH     »O    O3  Th<    O3    'rfl    00    CO    00    C^J 

tO    Tf    ^    CO    CO  to    HH    HH    CO    CO    CM  to    «0    -*    r*    CO    &  tO    T£    3<    CO    CO    CM    OJ 

O3    CO    O    CO    00    (N  b*    ^    ^H    CO    TH    tO  b-    CO    tO    CO    O3    tO  O3    ^H    ^    tO    b-    tO    Tt< 

£8  S3  £8  i^_S^_a^  ^?^'^'^^  §^38  £  8  a 

CM    (M    O3   b«   IN    00  C<l    (N    CO   (N    O   b-  to   tO   b-   b-   CO    CO  OO    IO   O   to   O3    IH    CO 

tOO^OlTfioO  COO3lOTHb»CQ  CMb-C^b^C^b-  b»COO3rfO5lod 

tOtOr^cOCOlN  ^COCOCO<N(N  tO^^cOCOC^  ^^COCO^CM^ 

oo  p  p  03  oo  to  co  co  O3COCOCO  c^cooir-icocs  co  i— i  03  co  co  oo  c^ 


^^C^^r      QCOC^CTSIOTH      c?ir3OCOT-ico     tOTHcorv3fyi«->o3 


re 


OcOC^O3tOTH      OtoOcOT-tc£ 
•^COCO<NCM(N       tO^-^COCOC^ 


^JS^SS^*-5  '— itOO3COtOb-  tOb-OCNrfi    ^  00    T-I    10   OO    O    i— (    'M 

CMOOTtiOCOiN  OOTt*Ob-COO3  O3tO(MOOTjHO  OOtOTHb-^OO 

Tt<COCOCOCM(N  COCOCOlNfN'H  COCOCOCM(N(N  COCOCo'cMCMCMTH 

J^'^JJOb-OOOO  CO    CO    r- 1    00    TJH    O  tOOtOCSCMtO  C^OOr^OSTfiOOC^ 

00(MOOrJHp  ^    TH    00   ^    ^    00  b-rjnpCOC003  CO^OtO^OoS 

•^   CO   CO   C<1    CM    CM  CO   CO   (N   (N   (N"   IH  CO   CO   Co'   CM*   <N'   TH  CO   CO   CM   CM    CM'   TH    TH 

i^§F3c^ot  2So^^oT^§  S^SIo^g  c^c5b3S2^^ 

cocococMc^iH  cocMcMc^THiH  cococ^c^c^iH  COCO'C^'C^'CN'TH'TH' 

^S5!^^S  !NtOGCOCN1CO  C^IOOOCOC^OO  OOOOOOOOOCOr^ 

COC^O3COCMOO  O3COCOiHOOtO  Tt<T-Hb<-TtiTHb>«  THOOtOCMO3COCO 

CO  CO  CM  CM   (M   TH  (N   (N   CM   <N   TH  TH  CO  CO  (M   (N   CM   TH  CO  CM'  CM*  C<J  TH  TH   TH 

i2   ^    !T*    00    r^    O  O    to   TH    10   O»   CM  00   b-   b»   CO    Tt<    T-H  O3    (N    TJH    CO   b-   b-   b*. 

TTi-«OO'<TTHOC  b*^(MO3CO^  CNO3COCOOt>  O3b»^THOOtOCM 

CO  CO  CM   CM   CM   TH  (N   CM   CM   TH   ^H  TH  CO  CM   CM   CM   CM'  TH  (N  CM'  CM'  CM'  i-i  TH   TH 

copcocopb^  Sc^pooSco  TH«Joc^pco  oo^ocoob^^^ 

CO   CO   CM   CM   CM    TH  (N    CM   CM    TH   TH   TH  CO   CM   CM   CM'   TH    TH  (N   CM   CM   CM   TH   TH   ^-1 

p    °0_   b-  b-   p    to   rf    CO    CM  p    to   Tt<    CO   CM    TH  (N    TH    O    O   O3   00   b- 

CO   CO   CO    *O   IO    ^O  ^^    C^    C^    ^1    ^3    r*Q  if5   \r\    \(*\   10   10    10 

QOQOQOQOQOQO  OOQOQOQOQOOO  OOQOOO&OOOoO  QOQOOOQOQOQOQO 

tOtO^TjHCOCO  pop    pOO  •^rt*COCOCM(N  tOT^rt<COCOCMCM 

SoSoSoSoSoSo  g  g  g  g  g  g  sg^oi^so'so  gggsgggg 

COCOCOCOCOCO  COCOt^-00030  b>»b»COcOCOcO  ^TtiTtir^tototO 

^  ^  £:  ^  N!  ^  000^^8  £:£:£:£:£:£:  g  g  g  g  g  g  g 

P   P    P   p    «O    CO  CO   TfH    to   CO   b-   00  CM    CM    CM    CO    CO   CO  O3    O3   O   O   TH    TH    CM 


THC^CNCOCOTf!       COb»O3OCMCO       TH,— IC^C^ICOCO       O3OTHCMCOH^iO 

gggggg         g^^ggg        gggg'gg         g'cggg^gg 

P^C^CMCOCO       COOOOCMCOtO       CM    (N    CO    CO    HH    IO       to    CO    b»    00    O3    O    iH 


CO 


64 


TABLES 


J 


bC 


8  * 


8  6 

a  fl 

•»  §  I 

S             O  «*H 

•S     w  ^ 


•fj       ^ 
0)        03 

8     ^ 


s 


II 


§    "i 

»  w 


0 

o 
o 

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-\ 

0 
0 

ct  ^_ 


l 


O5  O  00  OS  t^ 

T-H  CO  r—  1  CO  i—  1 

TH  CO  CO  (M  <N 


CO 


<N 
(N 


Oi   CO   GO    O  to   t>   Oi   C5 

CO 


o  to  o  to  o 

CO  <M  <N 


(M  l>  CO  CO  CX) 


i-  1>  <M  00 
CO  (N  <M  rH 


rH    CO    CO    Tt<    CO  CO    rH    GO    CO    Oi 


O5    CO    I>-    O     rH  IO 


OS  CD  C^  Oi  (M 

GO  iO  C^  GO  iO 
<M  <M  C^J  I-H  T-H 


O5  rH  CO  CO  CO 


<N  T-H  05  00  CO 
CD  CO  O5  CO  CO 

(N   (M   T-H   T-H   T-H 


GO  CO  rt<  (N  O> 

CO  CO  O  !>•  CO 


CO  CO  CO  <N  rH 

tO  (M  Oi  CO  CO 

(M   (N   T-H   T-H   T-i 


(M  t^  O  CO  CO 
<M  O5  1>  "*  T-H 


T-H   T-H   T-H   T-H 


T-H  GO  »O  (M 


t^  rH  TfH  CO  GO 
(N  O  i>  -^  rH 


(M   T-H   T-H   T-H   T-H 


rH  O  Oi  GO  >O 
GO  CO  CO  T-H  O5 

T-H  T—  1  T—  1  rH  O 


GO  CO  GO 
tO  CO  O 


d 

GO 


o 

GO  GO  GO  GO 


(M  (N  <M  <N  (M 

GO  GO  GO  GO  GO 


0  0  0  0  T-H 


GO  1>  1>  CO  CO 


CO  CO  CO 

CO  CO  CO 


O5  O5  O  O  rH 


o 

CO  CO  1^  l>  !>• 


10  1>  05  O 

Oi  O^  O^  CO 

»o  10  »o  10 


l>    Oi     rH     CO    tO 
ggg 


iO  CO  l>  l>  GO 

CO   cO    CO   CO   CO 
CO   cO   cO   co   cO 


«o  co  i>  GO  o> 

CO    CO    CO    CO    CO 
CO   CO   CO   CO   CO 

|<o  to 

r-i|M      |H  n|oo     IH  „  * 


TABLES 


65 


O    CO 


CO   <M    05      rH 


b~   (N   b-   rH 

CO    CO    C^    (N 


IH  10  oo    to  i-i  p  oq  p 

8(NrH      CO   CO   <N    <N   <N 


W   (N   W   (N  05 

rJH    O   CO   tN   1> 

CO    CO    <N    Oa    rH 


CO    O    tO    O5    rH 


00     TfH     O5 


88 


<M    0 

co  co 


O    b-    ^         rH     O    00    CO    O5 

N 


O    rH    10     CD     00 


^82 


00    Oi    l> 
O5    CO    CO 


CO    rH    O5 
IO    rH    CO 


CO    rH    00    (M        CD    O5 


O5    t^       .05 


O5         C^     00     IO     rH 
rH         CO     CN     (N     (N 


t>.         TjH     O     CO     rH 

rH         CO     CO     CM     <N 


O  CO  b- 

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THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  5O  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $I.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


29  1936 


LD  21-100m-8,'34 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


^S?.v.^'  •"£.";•_ 


